_Doctrine of Infinite Quantities—Labours of Pappus—Kepler—Cavaleri—
Roberval—Fermat—Wallis—Newton discovers the Binomial Theorem—and
the Doctrine of Fluxions in 1666—His Manuscript Work containing
this Doctrine communicated to his Friends—His Treatise on Fluxions—
His Mathematical Tracts—His Universal Arithmetic—His Methodus
Differentialis—His Geometria Analytica—His Solution of the Problems
proposed by Bernouilli and Leibnitz—Account of the celebrated
Dispute respecting the Invention of Fluxions—Commercium Epistolicum—
Report of the Royal Society—General View of the Controversy._
Previous to the time of Newton, the doctrine of infinite quantities had
been the subject of profound study. The ancients made the first step in
this curious inquiry by a rude though ingenious attempt to determine
the area of curves. The method of exhaustions which was used for this
purpose consisted in finding a given rectilineal area to which the
inscribed and circumscribed polygonal figures continually approached
by increasing the number of their sides. This area was obviously the
area of the curve, and in the case of the parabola it was found by
Archimedes to be two-thirds of the area formed by multiplying the
ordinate by the abscissa. Although the synthetical demonstration of the
results was perfectly conclusive, yet the method itself was limited and
imperfect.
The celebrated Pappus of Alexandria followed Archimedes in the same
inquiries; and in his demonstration of the property of the centre of
gravity of a plane figure, by which we may determine the solid formed
by its revolution, he has shadowed forth the discoveries of later times.
In his curious tract on Stereometry, published in 1615, Kepler made
some advances in the doctrine of infinitesimals. Prompted to the task
by a dispute with the seller of some casks of wine, he studied the
measurement of solids formed by the revolution of a curve round any
line whatever. In solving some of the simplest of these problems, he
conceived a circle to be formed of an infinite number of triangles
having all their vertices in the centre, and their infinitely small
bases in the circumference of the circle, and by thus rendering
familiar the idea of quantities infinitely great and infinitely small,
he gave an impulse to this branch of mathematics. The failure of
Kepler, too, in solving some of the more difficult of the problems
which he himself proposed roused the attention of geometers, and seems
particularly to have attracted the notice of Cavaleri.
This ingenious mathematician was born at Milan in 1598, and was
Professor of Geometry at Bologna. In his method of Indivisibles, which
was published in 1635, he considered a line as composed of an infinite
number of points, a surface of an infinite number of lines, and a solid
of an infinite number of surfaces; and he lays it down as an axiom
that the infinite sums of such lines and surfaces have the same ratio
when compared with the linear or superficial unit, as the surfaces and
solids which are to be determined. As it is not true that an infinite
number of infinitely small points can make a line, or an infinite
number of infinitely small lines a surface, Pascal removed this verbal
difficulty by considering a line as composed of an infinite number of
infinitely short lines, a surface as composed of an infinite number of
infinitely narrow parallelograms, and a solid of an infinite number
of infinitely thin solids. But, independent of this correction, the
conclusions deduced by Cavaleri are rigorously true, and his method of
ascertaining the ratios of areas and solids to one another, and the
theorems which he deduced from it may be considered as forming an era
in mathematics.
By the application of this method, Roberval and Toricelli showed that
the area of the cycloid is three times that of its generating circle,
and the former extended the method of Cavaleri to the case where the
powers of the terms of the arithmetical progression to be summed were
fractional.
In applying the doctrine of infinitely small quantities to determine
the tangents of curves, and the maxima and minima of their ordinates,
both Roberval and Fermat made a near approach to the invention of
fluxions—so near indeed that both Lagrange and Laplace[56] have
pronounced the latter to be the true inventer of the differential
calculus. Roberval supposed the point which describes a curve to be
actuated by two motions, by the composition of which it moves in the
direction of a tangent; and had he possessed the method of fluxions, he
could, in every case, have determined the relative velocities of these
motions, which depend on the nature of the curve, and consequently the
direction of the tangent which he assumed to be in the diagonal of a
parallelogram whose sides had the same ratio as the velocities. But as
he was able to determine these velocities only in the conic sections,
&c. his ingenious method had but few applications.
The labours of Peter Fermat, a counsellor of the parliament of
Toulouse, approached still nearer to the fluxionary calculus. In his
method of determining the maxima and minima of the ordinates of curves,
he substitutes _x_ + _e_ for the independent variable _x_ in the
function which is to become a maximum, and as these two expressions
should be equal when _e_ becomes infinitely small or 0, he frees this
equation from surds and radicals, and after dividing the whole by _e_,
_e_ is made = 0, and the equation for the maximum is thus obtained.
Upon a similar principle he founded his method of drawing tangents to
curves. But though the methods thus used by Fermat are in principle
the same with those which connect the theory of tangents and of maxima
and minima with the analytical method of exhibiting the differential
calculus, yet it is a singular example of national partiality to
consider the inventer of these methods as the inventer of the method of
fluxions.
“One might be led,” says Mr. Herschel, “to suppose by Laplace’s
expression that the calculus of finite differences had then already
assumed a systematic form, and that Fermat had actually observed
the relation between the two calculi, and derived the one from the
other. The latter conclusion would scarcely be less correct than the
former. No method can justly be regarded as bearing any analogy to the
differential calculus which does not lay down a system of rules (no
matter on what considerations founded, by what names called, or by
what extraneous matter enveloped) by means of which the second term of
the development of any function of _x_ + _e_ in powers of _e_, can be
correctly calculated, ‘quæ extendet se,’ to use Newton’s expression,
‘_citra_ ullum molestum calculum in terminis surdis æque ac in integris
procedens.’ It would be strange to suppose Fermat or any other in
possession of such a method before any single surd quantity had ever
been developed in a series. But, in point of fact, his writings present
no trace of the kind; and this, though fatal to his claim, is allowed
by both the geometers cited. Hear Lagrange’s candid avowal. ‘Il fait
disparaitre dans cette equation,’ that of the maximum between _x_
and _e_, ‘les radicaux et les fractions s’il y en à.’ Laplace, too,
declares that ‘il savoit etendre son calcul aux fonctions irrationelles
en se debarrassant des irrationalités par l’elevation des radicaux
aux puissances.’ This is at once giving up the point in question. It
is allowing unequivocally that Fermat in these processes only took a
circuitous route to avoid a difficulty which it is one of the most
express objects of the differential calculus to face and surmount.
The whole claim of the French geometer arises from a confusion (too
often made) of the calculus and its applications, the means and the
end, under the sweeping head of ‘nouveaux calculs’ on the one hand,
and an assertion somewhat too unqualified, advanced in the warmth and
generality of a preface, on the other.”[57]
The discoveries of Fermat were improved and simplified by Hudde,
Huygens, and Barrow; and by the publication of the _Arithmetic of
Infinites_ by Dr. Wallis, Savilian professor of geometry at Oxford,
mathematicians were conducted to the very entrance of a new and
untrodden field of discovery. This distinguished author had effected
the quadrature of all curves whose ordinates can be expressed by any
direct integral powers; and though he had extended his conclusions
to the cases where the ordinates are expressed by the inverse or
fractional powers, yet he failed in its application. Nicolas Mercator
(Kauffman) surmounted the difficulty by which Wallis had been baffled,
by the continued division of the numerator by the denominator to
infinity, and then applying Wallis’s method to the resulting positive
powers. In this way he obtained, in 1667, the first general quadrature
of the hyperbola, and, at the same time, gave the regular development
of a function in series.
In order to obtain the quadrature of the circle, Dr. Wallis considered
that if the equations of the curves of which he had given the
quadrature were arranged in a series, beginning with the most simple,
these areas would form another series. He saw also that the equation of
the circle was intermediate between the first and second terms of the
first series, or between the equation of a straight line and that of a
parabola, and hence he concluded, that by interpolating a term between
the first and second term of the second series, he would obtain the
area of the circle. In pursuing this singularly beautiful thought, Dr.
Wallis did not succeed in obtaining the indefinite quadrature of the
circle, because he did not employ general exponents; but he was led to
express the entire area of the circle by a fraction, the numerator and
denominator of which are each obtained by the continued multiplication
of a certain series of numbers.
Such was the state of this branch of mathematical science, when
Newton, at an early age, directed to it the vigour of his mind. At
the very beginning of his mathematical studies, when the works of
Dr. Wallis fell into his hands, he was led to consider how he could
interpolate the general values of the areas in the second series of
that mathematician. With this view he investigated the arithmetical
law of the coefficients of the series, and obtained a general method
of interpolating, not only the series above referred to, but also
other series. These were the first steps taken by Newton, and, as he
himself informs us, they would have entirely escaped from his memory
if he had not, a few weeks before,[58] found the notes which he made
upon the subject. When he had obtained this method, it occurred to him
that the very same process was applicable to the ordinates, and, by
following out this idea, he discovered the general method of reducing
radical quantities composed of several terms into infinite series, and
was thus led to the discovery of the celebrated _Binomial Theorem_. He
now neglected entirely his methods of interpolation, and employed that
theorem alone as the easiest and most direct method for the quadratures
of curves, and in the solution of many questions which had not even
been attempted by the most skilful mathematicians.
After having applied the Binomial theorem to the rectification of
curves, and to the determination of the surfaces and contents of
solids, and the position of their centres of gravity, he discovered the
general principle of deducing the areas of curves from the ordinate,
by considering the area as a nascent quantity, increasing by continual
fluxion in the proportion of the length of the ordinate, and supposing
the abscissa to increase uniformly in proportion to the time. In
imitation of Cavalerius, he called the momentary increment of a line a
point, though it is not a geometrical point, but an infinitely short
line; and the momentary increment of an area or surface he called a
line, though it is not a geometrical line, but an infinitely narrow
surface. By thus regarding lines as generated by the motion of points,
surfaces by the motions of lines, and solids by the motion of surfaces,
and by considering that the ordinates, abscissæ, &c. of curves thus
formed, vary according to a regular law depending on the equation of
the curve, he deduces from this equation the velocities with which
these quantities are generated; and by the rules of infinite series he
obtains the ultimate value of the quantity required. To the velocities
with which every line or quantity is generated, Newton gave the name
of _Fluxions_, and to the lines or quantities themselves that of
_Fluents_. This method constitutes the doctrine of fluxions which
Newton had invented previous to 1666, when the breaking out of the
plague at Cambridge drove him from that city, and turned his attention
to other subjects.
But though Newton had not communicated this great invention to any
of his friends, he composed his treatise, entitled _Analysis per
equationes numero terminorum infinitas_, in which the principle of
fluxions and its numerous applications are clearly pointed out. In
the month of June, 1669, he communicated this work to Dr. Barrow, who
mentions it in a letter to Mr. Collins, dated the 20th June, 1669, as
the production of a friend of his residing at Cambridge, who possesses
a fine genius for such inquiries. On the 31st July, he transmitted the
work to Collins; and having received his approbation of it, he informs
him that the name of the author of it was Newton, a fellow of his
own college, and a young man who had only two years before taken his
degree of M.A. Collins took a copy of this treatise, and returned the
original to Dr. Barrow; and this copy having been found among Collins’s
papers by his friend Mr. William Jones, and compared with the original
manuscript borrowed from Newton, it was published with the consent of
Newton in 1711, nearly fifty years after it was written.
Though the discoveries contained in this treatise were not at
first given to the world, yet they were made generally known to
mathematicians by the correspondence of Collins, who communicated them
to James Gregory; to MM. Bertet and Vernon in France; to Slusius in
Holland; to Borelli in Italy; and to Strode, Townsend, and Oldenburg,
in letters dated between 1669 and 1672.
Hitherto the method of fluxions was known only to the friends of
Newton and their correspondents; but, in the first edition of the
Principia, which appeared in 1687, he published, for the first time,
the fundamental principle of the fluxionary calculus, in the second
lemma of the second book. No information, however, is here given
respecting the algorithm or notation of the calculus; and it was not
till 1693–5[?] that it was communicated to the mathematical world in
the second volume of Dr. Wallis’s works, which were published in that
year. This information was extracted from two letters of Newton written
in 1692.
About the year 1672, Newton had undertaken to publish an edition of
Kinckhuysen’s Algebra, with notes and additions. He therefore drew up
a treatise, entitled, _A Method of Fluxions_, which he proposed as an
introduction to that work; but the fear of being involved in disputes
about this new discovery, or perhaps the wish to render it more
complete, or to have the sole advantage of employing it in his physical
researches, induced him to abandon this design. At a later period of
his life he again resolved to give it to the world; but it did not
appear till after his death, when it was translated into English, and
published in 1736, with a commentary by Mr. John Colson, Professor of
Mathematics in Cambridge.[59]
To the first edition of Newton’s Optics, which appeared in 1704, there
were added two mathematical treatises, entitled, _Tractatus duo de
speciebus et magnitudine figurarum curvilinearum_, the one bearing the
title of _Tractatus de Quadratura Curvarum_, and the other _Enumeratio
linearum tertii ordinis_. The first contains an explanation of the
doctrine of fluxions, and of its application to the quadrature of
curves; and the second a classification of seventy-two curves of the
third order, with an account of their properties. The reason for
publishing these two tracts in his Optics (in the subsequent editions
of which they are omitted) is thus stated in the advertisement:—“In
a letter written to M. Leibnitz in the year 1679, and published by
Dr. Wallis, I mentioned a method by which I had found some general
theorems about squaring curvilinear figures on comparing them with the
conic sections, or other the simplest figures with which they might be
compared. And some years ago I lent out a manuscript containing such
theorems; and having since met with some things copied out of it, I
have on this occasion made it public, prefixing to it an introduction,
and joining a scholium concerning that method. And I have joined with
it another small tract concerning the curvilineal figures of the second
kind, which was also written many years ago, and made known to some
friends, who have solicited the making it public.”
In the year 1707, Mr. Whiston published the algebraical lectures
which Newton had, during nine years, delivered at Cambridge, under
the title of _Arithmetica Universalis, sive de Compositione et
Resolutione Arithmetica Liber_. We are not accurately informed how
Mr. Whiston obtained possession of this work; but it is stated by one
of the editors of the English edition, that “Mr. Whiston thinking it
a pity that so noble and useful a work should be doomed to a college
confinement, obtained leave to make it public.” It was soon afterward
translated into English by Mr. Ralphson; and a second edition of it,
with improvements by the author, was published at London in 1712,
by Dr. Machin, secretary to the Royal Society. With the view of
stimulating mathematicians to write annotations on this admirable work,
the celebrated S’Gravesande published a tract, entitled, _Specimen
Commentarii in Arithmeticam Universalem_; and Maclaurin’s Algebra seems
to have been drawn up in consequence of this appeal.
Among the mathematical works of Newton we must not omit to enumerate a
small tract entitled, _Methodus Differentialis_, which was published
with his consent in 1711. It consists of six propositions, which
contain a method of drawing a parabolic curve through any given
number of points, and which are useful for constructing tables by the
interpolation of series, and for solving problems depending on the
quadrature of curves.
Another mathematical treatise of Newton’s was published for the
first time in 1779, in Dr. Horsley’s edition of his works.[60] It is
entitled, _Artis Analyticæ Specimina, vel Geometria Analytica_. In
editing this work, which occupies about 130 quarto pages, Dr. Horsley
used three manuscripts, one of which was in the handwriting of the
author; another, written in an unknown hand, was given by Mr. William
Jones to the Honourable Charles Cavendish; and a third, copied from
this by Mr. James Wilson, the editor of Robins’s works, was given to
Dr. Horsley by Mr. John Nourse, bookseller to the king. Dr. Horsley
has divided it into twelve chapters, which treat of infinite series;
of the reduction of affected equations; of the specious resolution
of equations; of the doctrine of fluxions; of maxima and minima;
of drawing tangents to curves; of the radius of curvature; of the
quadrature of curves; of the area of curves which are comparable with
the conic sections; of the construction of mechanical problems, and on
finding the lengths of curves.
In enumerating the mathematical works of our author, we must not
overlook his solutions of the celebrated problems proposed by
Bernouilli and Leibnitz. On the Kalends of January, 1697, John
Bernouilli addressed a letter to the most distinguished mathematicians
in Europe,[61] challenging them to solve the two following problems:
1. To determine the curve line connecting two given points which are
at different distances from the horizon, and not in the same vertical
line, along which a body passing by its own gravity, and beginning
to move at the upper point, shall descend to the lower point in the
shortest time possible.
2. To find a curve line of this property that the two segments of a
right line drawn from a given point through the curve, being raised to
any given power, and taken together, may make every where the same sum.
On the day after he received these problems, Newton addressed to Mr.
Charles Montague, the President of the Royal Society, a solution of
them both. He announced that the curve required in the first problem
must be a cycloid, and he gave a method of determining it. He solved
also the second problem, and he showed that by the same method other
curves might be found which shall cut off three or more segments having
the like properties. Leibnitz, who was struck with the beauty of the
problem, requested Bernouilli, who had allowed six months for its
solution, to extend the period to twelve months. This delay was readily
granted, solutions were obtained from Newton, Leibnitz, and the Marquis
de L’Hopital; and although that of Newton was anonymous, yet Bernouilli
recognised in it his powerful mind, “_tanquam_,” says he, “_ex ungue
leonem_,” as the lion is known by his claw.
The last mathematical effort of our author was made with his usual
success, in solving a problem which Leibnitz proposed in 1716, in a
letter to the Abbé Conti, “for the purpose, as he expressed it, of
feeling the pulse of the English analysts.” The object of this problem
was to determine the curve which should cut at right angles an infinity
of curves of a given nature, but expressible by the same equation.
Newton received this problem about five o’clock in the afternoon, as
he was returning from the Mint; and though the problem was extremely
difficult, and he himself much fatigued with business, yet he finished
the solution of it before he went to bed.
Such is a brief account of the mathematical writings of Sir Isaac
Newton, not one of which were voluntarily communicated to the world by
himself. The publication of his Universal Arithmetic is said to have
been a breach of confidence on the part of Whiston; and, however this
may be, it was an unfinished work, never designed for the public. The
publication of his Quadrature of Curves, and of his Enumeration of
Curve Lines, was rendered necessary, in consequence of plagiarisms from
the manuscripts of them which he had lent to his friends, and the rest
of his analytical writings did not appear till after his death. It is
not easy to penetrate into the motives by which this great man was on
these occasions actuated. If his object was to keep possession of his
discoveries till he had brought them to a higher degree of perfection,
we may approve of the propriety, though we cannot admire the prudence
of such a step. If he wished to retain to himself his own methods, in
order that he alone might have the advantage of them in prosecuting
his physical inquiries, we cannot reconcile so selfish a measure with
that openness and generosity of character which marked the whole of his
life. If he withheld his labours from the world in order to avoid the
disputes and contentions to which they might give rise, he adopted
the very worst method of securing his tranquillity. That this was the
leading motive under which he acted, there is little reason to doubt.
The early delay in the publication of his method of fluxions, after the
breaking out of the plague at Cambridge, was probably owing to his not
having completed the algorithm of that calculus; but no apology can be
made for the imprudence of withholding it any longer from the public.
Had he published this noble discovery even previous to 1673, when his
great rival had not even entered upon those studies which led him to
the same method, he would have secured to himself the undivided honour
of the invention, and Leibnitz could have aspired to no other fame but
that of an improver of the doctrine of fluxions. But he unfortunately
acted otherwise. He announced to his friends that he possessed a method
of great generality and power; he communicated to them a general
account of its principles and applications; and the information which
was thus conveyed directed the attention of mathematicians to subjects
to which they might not have otherwise applied their powers. In this
way the discoveries which he had previously made were made subsequently
by others; and Leibnitz, in place of appearing in the theatre of
science as the disciple and the follower of Newton, stood forth with
all the dignity of a rival; and, by the early publication of his
discoveries had nearly placed himself on the throne which Newton was
destined to ascend.
It would be inconsistent with the popular nature of a work like this,
to enter into a detailed history of the dispute between Newton and
Leibnitz respecting the invention of fluxions. A brief and general
account of it, however, is indispensable.
In the beginning of 1673, Leibnitz came to London in the suite of
the Duke of Hanover, and he became acquainted with the great men who
then adorned the capital of England. Among these was Oldenburg, a
countryman of his own, who was then secretary to the Royal Society.
About the beginning of March, in the same year, Leibnitz went to
Paris, where, with the assistance of Huygens, he devoted himself to
the study of the higher geometry. In the month of July he renewed his
correspondence with Oldenburg, and he communicated to him some of
the discoveries which he had made relative to series, particularly
the series for a circular arc in terms of the tangent. Oldenburg
informed him in return of the discoveries on series which had been
made by Newton and Gregory; and in 1676 Newton communicated to him,
through Oldenburg, a letter of fifteen closely printed quarto pages,
containing many of his analytical discoveries, and stating that he
possessed a general method of drawing tangents, which he thought it
necessary to conceal in two sentences of transposed characters. In this
letter neither the method of fluxions nor any of its principles are
communicated; but the superiority of the method over all others is so
fully described, that Leibnitz could scarcely fail to discover that
Newton possessed that secret of which geometers had so long been in
quest.
Had Leibnitz at the time of receiving this letter been entirely
ignorant of his own differential method, the information thus conveyed
to him by Newton could not fail to stimulate his curiosity, and excite
his mightiest efforts to obtain possession of so great a secret. That
this new method was intimately connected with the subject of series
was clearly indicated by Newton; and as Leibnitz was deeply versed in
this branch of analysis, it is far from improbable that a mind of such
strength and acuteness might attain his object by direct investigation.
That this was the case may be inferred from his letter to Oldenburg (to
be communicated to Newton) of the 21st June, 1677, where he mentions
that he had for some time been in possession of a method of drawing
tangents more general than that of Slusius, namely, by the differences
of ordinates. He then proceeds with the utmost frankness to explain
this method, which was no other than the differential calculus.
He describes the algorithm which he had adopted, the formation of
differential equations, and the application of the calculus to various
geometrical and analytical questions. No answer seems to have been
returned to this letter either by Newton or Oldenburg, and, with the
exception of a short letter from Leibnitz to Oldenburg, dated 12th
July, 1677, no further correspondence seems to have taken place. This,
no doubt, arose from the death of Oldenburg in the month of August,
1677,[62] when the two rival geometers pursued their researches
with all the ardour which the greatness of the subject was so well
calculated to inspire.
In the hands of Leibnitz the differential calculus made rapid progress.
In the _Acta Eruditorum_, which was published at Leipsic in November,
1684, he gave the first account of it, describing its algorithm in the
same manner as he had done in his letter to Oldenburg, and pointing
out its application to the drawing of tangents, and the determination
of maxima and minima. He makes a remote reference to the _similar_
calculus of Newton, but lays no claim to the sole invention of the
differential method. In the same work for June, 1686, he resumes
the subject; and when Newton had not published a single word upon
fluxions, and had not even made known his notation, the differential
calculus was making rapid advances on the Continent, and in the hands
of James and John Bernouilli had proved the means of solving some of
the most important and difficult problems.
The silence of Newton was at last broken, and in the second lemma
of the second book of the Principia, he explained the fundamental
principle of the fluxionary calculus. His explanation, which occupied
only three pages, was terminated with the following scholium:—“In a
correspondence which took place about ten years ago between that very
skilful geometer, G. G. Leibnitz, and myself, I announced to him that
I possessed a method of determining maxima and minima, of drawing
tangents, and of performing similar operations which was equally
applicable to rational and irrational quantities, and concealed the
same in transposed letters involving this sentence, (_data equatione
quotcunque fluentes quantitates involvente, fluxiones invenire et
vice versa_). This illustrious man replied that he also had fallen
on a method of the same kind, and he communicated to me his method
which scarcely differed from mine except in the notation [and in the
idea of the generation of quantities.”][63] This celebrated scholium,
which is so often referred to in the present controversy, has, in
our opinion, been much misapprehended. While M. Biot considers it
as “eternalizing the rights of Leibnitz by recognising them in the
Principia,” Professor Playfair regards it as containing “a highly
favourable opinion on the subject of the discoveries of Leibnitz.”
To us it appears to be nothing more than the simple statement of the
fact, that the method communicated by Leibnitz was nearly the same as
his own; and this much he might have said, whether he believed that
Leibnitz had seen the fluxionary calculus among the papers of Collins,
or was the independent inventor of his own. It is more than probable,
indeed, that when Newton wrote this scholium he regarded Leibnitz as a
second inventor; but when he found that Leibnitz and his friends had
showed a willingness to believe, and had even ventured to throw out the
suspicion, that he himself had borrowed the doctrine of fluxions from
the differential calculus, he seems to have altered the opinion which
he had formed of his rival, and to have been willing in his turn to
retort the charge.
This change of opinion was brought about by a series of circumstances
over which he had no control. M. Nicolas Fatio de Duillier, a Swiss
mathematician, resident in London, communicated to the Royal Society,
in 1699, a paper on the line of quickest descent, which contains
the following observations:—“Compelled by the evidence of facts, I
hold Newton to have been the first inventor of this calculus, and
the earliest by several years; and whether Leibnitz, the _second
inventor_, has borrowed any thing from the other, I would prefer to my
own judgment that of those who have seen the letters and other copies
of the same manuscripts of Newton.” This imprudent remark, which by
no means amounts to a charge of plagiarism, for Leibnitz is actually
designated the _second inventor_, may be considered as showing that
the English mathematicians had been cherishing suspicions unfavourable
to Leibnitz, and there can be no doubt that a feeling had long
prevailed that this mathematician either had, or might have seen, among
the papers of Collins, the “_Analysis per Equationes, &c._,” which
contained the principles of the fluxionary method. Leibnitz replied
to the remark of Duillier with much good feeling. He appealed to the
facts as exhibited in his correspondence with Oldenburg; he referred to
Newton’s scholium as a testimony in his favour; and, without disputing
or acknowledging the priority of Newton’s claim, he asserted his own
right to the invention of the differential calculus. Fatio transmitted
a reply to the Leipsic Acts; but the editor refused to insert it. The
dispute, therefore, terminated, and the feelings of the contending
parties continued for some time in a state of repose, though ready to
break out on the slightest provocation.
When Newton’s Optics appeared in 1704, accompanied by his Treatise on
the Quadrature of Curves, and his enumeration of lines of the third
order, the editor of the Leipsic Acts (whom Newton supposed to be
Leibnitz himself) took occasion to review the first of these tracts.
After giving an imperfect analysis of its contents, he compared the
method of fluxions with the differential calculus, and, in a sentence
of some ambiguity, he states that Newton employed fluxions in place
of the differences of Leibnitz, and made use of them in his Principia
in the same manner as Honoratus Fabri, in his Synopsis of Geometry,
had substituted progressive motion in place of the indivisibles of
Cavaleri.[64] As Fabri, therefore, was not the inventor of the method
which is here referred to, but borrowed it from Cavaleri, and only
changed the mode of its expression, there can be no doubt that the
artful insinuation contained in the above passage was intended to
convey the impression that Newton had _stolen_ his method of fluxions
from Leibnitz. The indirect character of this attack, in place
of mitigating its severity, renders it doubly odious; and we are
persuaded that no candid reader can peruse the passage without a strong
conviction that it justifies, in the fullest manner, the indignant
feelings which it excited among the English philosophers. If Leibnitz
was the author of the review, or if he was in any way a party to it, he
merited the full measure of rebuke which was dealt out to him by the
friends of Newton, and deserved those severe reprisals which doubtless
imbittered the rest of his days. He who dared to accuse a man like
Newton, or indeed any man holding a fair character in society, with
the odious crime of plagiarism, placed himself without the pale of
the ordinary courtesies of life, and deserved to have the same charge
thrown back upon himself. The man who conceives his fellow to be
capable of such intellectual felony, avows the possibility of himself
committing it, and almost substantiates the weakest evidence of the
worst accusers.
Dr. Keill, as the representative of Newton’s friends, could not brook
this base attack upon his countryman. In a letter printed in the
Philosophical Transactions for 1708, he maintained that Newton was
“beyond all doubt” the first inventor of fluxions. He referred for
a direct proof of this to his letters published by Wallis; and he
asserted “that the same calculus was afterward published by Leibnitz,
the name and the mode of notation being changed.” If the reader is
disposed to consider this passage as retorting the charge of plagiarism
upon Leibnitz, he will readily admit that the mode of its expression
is neither so coarse nor so insidious as that which is used by the
writer in the Leipsic Acts. In a letter to Hans Sloane, dated March,
1711, Leibnitz complained to the Royal Society of the treatment he had
received. He expressed his conviction that Keill had erred more from
rashness of judgment than from any improper motive, and that he did not
regard the accusation as a calumny; and he requested that the society
would oblige Mr. Keill to disown publicly the injurious sense which
his words might bear. When this letter was read to the society, Keill
justified himself to Sir Isaac Newton and the other members by showing
them the obnoxious review of the Quadrature of Curves in the Leipsic
Acts. They all agreed in attaching the same injurious meaning to the
passage which we formerly quoted, and authorized Keill to explain and
defend his statement. He accordingly addressed a letter to Sir Hans
Sloane, which was read at the society on the 24th May, 1711, and a copy
of which was ordered to be sent to Leibnitz. In this letter, which is
one of considerable length, he declares that he never meant to state
that Leibnitz knew either the name of Newton’s method or the form of
notation, and that the real meaning of the passage was, “that Newton
was the first inventor of fluxions or of the differential calculus,
and that he had given, in two letters to Oldenburg, and which he had
transmitted to Leibnitz, indications of it sufficiently intelligible to
an acute mind, from which Leibnitz derived, or at least might derive,
the principles of his calculus.”
The charge of plagiarism which Leibnitz thought was implied in the
former letter of his antagonist is here greatly modified, if not
altogether denied. Keill expresses only an _opinion_ that the letter
_seen_ by Leibnitz contained intelligible indications of the fluxionary
calculus. Even if this opinion were correct, it is no proof that
Leibnitz either saw these indications or availed himself of them, or if
he did perceive them, it might have been in consequence of his having
previously been in possession of the differential calculus, or having
enjoyed some distant view of it. Leibnitz should, therefore, have
allowed the dispute to terminate here; for no ingenuity on his part,
and no additional facts, could affect an opinion which any other person
as well as Keill was entitled to maintain.
Leibnitz, however, took a different view of the subject, and wrote a
letter to Sir Hans Sloane, dated December 19, 1711, which excited new
feelings, and involved him in new embarrassments. Insensible to the
mitigation which had been kindly impressed upon the supposed charge
against his honour, he alleges that Keill had attacked his candour and
sincerity more openly than before;—that he acted without any authority
from Sir Isaac Newton, who was the party interested;—and that it was
in vain to justify his proceedings by referring to the provocation
in the Leipsic Acts, because in that journal _no injustice had been
done to any party, but every one had received what was his due_. He
branded Keill with the odious appellation of an upstart, and one little
acquainted with the circumstances of the case;[65] he called upon
the society to silence his vain and unjust clamours,[66] which, he
believed, were disapproved by Newton himself, who was well acquainted
with the facts, and who, he was persuaded, would willingly give his
opinion on the matter.
This unfortunate letter was doubtless the cause of all the rancour and
controversy which so speedily followed, and it placed his antagonist
in a new and a more favourable position. It may be correct, though
few will admit it, that Keill’s second letter was more injurious than
the first; but it was not true that Keill acted without the authority
of Newton, because Keill’s letter was approved of and transmitted by
the Royal Society, of which Newton was the president, and therefore
became the act of that body. The obnoxious part, however, of Leibnitz’s
letter consisted in his appropriating to himself the opinions of the
reviewer in the Leipsic Acts, by declaring that, in a review which
charged Newton with plagiarism, every person had got what was his due.
The whole character of the controversy was now changed: Leibnitz places
himself in the position of the party who had first disturbed the
tranquillity of science by maligning its most distinguished ornament;
and the Royal Society was imperiously called upon to throw all the
light they could upon a transaction which had exposed their venerable
president to so false a charge. The society, too, had become a party to
the question, by their approbation and transmission of Keill’s second
letter, and were on that account alone bound to vindicate the step
which they had taken.
When the letter of Leibnitz, therefore, was read, Keill appealed to the
registers of the society for the proofs of what he had advanced; Sir
Isaac also expressed his displeasure at the obnoxious passage in the
Leipsic Review, and at the defence of it by Leibnitz, and he left it to
the society to act as they thought proper. A committee was therefore
appointed on the 11th March, consisting of Dr. Arbuthnot, Mr. Hill,
Dr. Halley, Mr. Jones, Mr. Machin, and Mr. Burnet, who were instructed
to examine the ancient registers of the society, to inquire into the
dispute, and to produce such documents as they should find, together
with their own opinions on the subject. On the 24th April the committee
produced the following report:—
“We have consulted the letters and letter-books in the custody of the
Royal Society, and those found among the papers of Mr. John Collins,
dated between the years 1669 and 1677, inclusive; and showed them to
such as knew and avouched the hands of Mr. Barrow, Mr. Collins, Mr.
Oldenburg, and Mr. Leibnitz; and compared those of Mr. Gregory with
one another, and with copies of some of them taken in the hand of
Mr. Collins; and have extracted from them what relates to the matter
referred to us; all which extracts herewith delivered to you we believe
to be genuine and authentic. And by these letters and papers we find,—
“I. Mr. Leibnitz 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 returned by London and Amsterdam to
Hanover: and that Mr. Collins was very free in communicating to able
mathematicians what he had received from Mr. Newton and Mr. Gregory.
“II. That when Mr. Leibnitz was the first time in London, he contended
for the invention of another differential method properly so called;
and, notwithstanding that he was shown by Dr. Pell that it was Newton’s
method, persisted in maintaining it to be his own invention, by reason
that he had found it by himself without knowing what Newton had done
before, and had much improved it. And we find no mention of his having
any other differential method than Newton’s before his letter of the
21st of June, 1677, which was a year after a copy of Mr. Newton’s
letter of the 10th of December, 1672, had been sent to Paris to be
communicated to him; and above four years after, Mr. Collins began
to communicate that letter to his correspondent; in which letter the
method of fluxions was sufficiently described to any intelligent person.
“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.
“IV. That the differential method is one and the same with the method
of fluxions, excepting the name and mode of notation; Mr. Leibnitz
calling those quantities differences which Mr. Newton calls moments or
fluxions; and marking them with the letter _d_—a mark not used by Mr.
Newton.
“And therefore we take the proper question to be not who invented this
or that method, but who was the first inventor of the method. And we
believe that those who have reputed Mr. Leibnitz the first inventor
knew little or nothing of his correspondence with Mr. Collins and Mr.
Oldenburg long before, nor of Mr. Newton’s having that method above
fifteen years before Mr. Leibnitz began to publish it in the Acta
Eruditorum of Leipsic.
“For which reason we reckon Mr. Newton the first inventor; and are
of opinion that Mr. Keill, in asserting the same, has been no ways
injurious to Mr. Leibnitz. And we submit to the judgment of the society
whether the extract and papers now presented to you, together with what
is extant to the same purpose in Dr. Wallis’s third volume, may not
deserve to be made public.”
This report being read, the society unanimously ordered the collection
of letters and manuscripts to be printed, and appointed Dr. Halley,
Mr. Jones, and Mr. Machin to superintend the press. Complete copies of
it, under the title of _Commercium Epistolicum D. Johannis Collins et
aliorum de analysi promota_, were laid before the society on the 8th
January, 1713, and Sir Isaac Newton, as president, ordered a copy to be
delivered to each person of the committee appointed for that purpose,
to examine it before its publication.
Leibnitz received information of the appearance of the Commercium
Epistolicum when he was at Vienna; and “being satisfied,” as he
expresses it, “that it must contain _malicious falsehoods_, I did not
think proper to send for it by post, but wrote to M. Bernouilli to give
me his sentiments. M. Bernouilli wrote me a letter dated at Basle,
June 7th, 1713, in which he said _that it appeared probable that Sir
Isaac Newton had formed his calculus after having seen mine_.”[67] This
letter was published by a friend of Leibnitz, with reflections, in a
loose sheet entitled _Charta Volans_, and dated July 29, 1713. It was
widely circulated without either the name of the author, printer, or
place of publication, and was communicated to the _Journal Literaire_
by another friend of Leibnitz, who added remarks of his own, and
stated, that when Newton published the Principia in 1687, _he did not
understand the true differential method; and that he took his fluxions
from Leibnitz_.
In this state of the controversy, Mr. Chamberlayne conceived the design
of reconciling the two distinguished philosophers; and in a letter
dated April 28, 1714,[68] he addressed himself to Leibnitz, who was
still at Vienna. In replying to this letter, Leibnitz declared that he
had given no occasion for the dispute; “that Newton procured a book to
be published, which was written purposely to discredit him, and sent
it to Germany, &c. as in the name of the society;” and he stated _that
there was room to doubt whether Newton knew his invention before he
had it of him_. Mr. Chamberlayne communicated this letter to Sir Isaac
Newton, who replied that Leibnitz had attacked his reputation in 1705,
by intimating that he had borrowed from him the method of fluxions;
that if Mr. C. could point out to him any thing in which he had injured
Mr. Leibnitz, he would give him satisfaction; that he would not retract
things which he knew to be true; and that he believed that the Royal
Society had done no injustice by the publication of the Commercium
Epistolicum.
The Royal Society, having learned that Leibnitz complained of their
having condemned him unheard, inserted a declaration in their journals
on the 20th May, 1714, that they did not pretend that the report
of their committee should pass for a decision of the society. Mr.
Chamberlayne sent a copy of this to Leibnitz, along with Sir Isaac’s
letter, and Dr. Keill’s answer to the papers inserted in the Journal
Literaire. After perusing these documents, M. Leibnitz replied, “that
Sir Isaac’s letter was written with very little civility; that he was
not in a humour to put himself in a passion against such people; that
there were other letters among those of Oldenburg and Collins which
should have been published; and that on his return to Hanover, he would
be able to publish a Commercium Epistolicum which would be of service
to the history of learning.” When this letter was read to the Royal
Society, Sir Isaac remarked, that the last part of it injuriously
accused the society of having made a partial selection of papers for
the Commercium Epistolicum; that he did not interfere in any way in
the publication of that work, and had even withheld from the committee
two letters, one from Leibnitz in 1693, and another from Wallis in
1695, which were highly favourable to his cause. He stated that he
did not think it right for M. Leibnitz himself, but that, if he had
letters to produce in his favour, that they might be published in the
Philosophical Transactions, or in Germany.
About this time the Abbé Conti, a noble Venetian, came to England. He
was a correspondent of Leibnitz, and in a letter which he had received
soon after his arrival,[69] he enters upon his dispute with Newton.
He charges the English “with wishing to pass for almost the only
inventors.” He declares “that Bernouilli had judged rightly in saying
that Newton did not possess before him the infinitesimal characteristic
and algorithm.” He remarks that Newton preceded him only in series; and
he confesses that during his second visit to England, “Collins showed
him part of his correspondence,” or, as he afterward expresses it, he
saw “some of the letters of Newton at Mr. Collins’s.” He then attacks
Sir Isaac’s philosophy, particularly his opinions about gravity and
vacuum, the intervention of God for the preservation of his creatures;
and he accuses him of reviving the occult qualities of the schools. But
the most remarkable passage in this letter is the following: “I am a
great friend of experimental philosophy, but Newton deviates much from
it _when he pretends that all matter is heavy_, or that each particle
of matter attracts every other particle.”
The above letter to the Abbé Conti was generally shown in London,
and came to be much talked of at court, in consequence of Leibnitz
having been privy counsellor to the Elector of Hanover when that
prince ascended the throne of England. Many persons of distinction,
and particularly the Abbé Conti, urged Newton to reply to Leibnitz’s
letter, but he resisted all their solicitations. One day, however, King
George I. inquired when Sir Isaac Newton’s answer to Leibnitz would
appear; and when Sir Isaac heard this, he addressed a long reply to the
Abbé Conti, dated February 26th, O. S. 1715–16. This letter, written
with dignified severity, is a triumphant refutation of the allegations
of his adversary; and the following passage deserves to be quoted, as
connected with that branch of the dispute which relates to Leibnitz’s
having seen part of Newton’s letters to Mr. Collins. “He complains of
the committee of the Royal Society, as if they had acted partially in
omitting what made against me; but he fails in proving the accusation.
For he instances in a paragraph concerning my ignorance, pretending
that they omitted it, and yet you will find it in the Commercium
Epistolicum, p. 547, lines 2, 3, and I am not ashamed of it. He saith
that he saw this paragraph in the hands of Mr. Collins when he was in
London the second time, that is in October, 1676. It is in my letter
of the 24th of October, 1676, and therefore he then saw that letter.
And in that and some other letters writ before that time, I described
my method of fluxions; and in the same letter I described also two
general methods of series, one of which is now claimed from me by Mr.
Leibnitz.” The letter concludes with the following paragraph: “But as
he has lately attacked me with an accusation which amounts to plagiary;
if he goes on to accuse me, it lies upon him by the laws of all
nations to prove his accusations, on pain of being accounted guilty of
calumny. He hath hitherto written letters to his correspondents full of
affirmations, complaints, and reflections, without proving any thing.
But he is the aggressor, and it lies upon him to prove the charge.”
In transmitting this letter to Leibnitz, the Abbé Conti informed him
that he himself had read with great attention, and without the least
prejudice, the Commercium Epistolicum, and the little piece[70] that
contains the extract; that he had also seen at the Royal Society the
original papers of the Commercium Epistolicum, and some other original
pieces relating to it. “From all this,” says he, “I infer, that, if
all the digressions are cut off, the only point is, whether Sir Isaac
Newton had the method of fluxions or infinitesimals before you, or
whether you had it before him. You published it first, it is true, but
you have owned also that Sir Isaac Newton had given many hints of it in
his letters to Mr. Oldenburg and others. This is proved very largely in
the Commercium, and in the extract of it. What answer do you give? This
is still wanting to the public, in order to form an exact judgment of
the affair.” The Abbé adds, that Mr. Leibnitz’s own friends waited for
his answer with great impatience, and that they thought he could not
dispense with answering, if not Dr. Keill, at least Sir Isaac Newton
himself, who had given him a defiance in express terms.
Leibnitz was not long in complying with this request. He addressed a
letter to the Abbé Conti, dated April 9th, 1716, but he sent it through
M. Ramond at Paris, to communicate it to others. When it was received
by the Abbé Conti, Newton wrote observations upon it, which were
communicated only to some of his friends, and which, while they placed
his defence on the most impregnable basis, at the same time threw much
light on the early history of his mathematical discoveries.
The death of Leibnitz on the 14th November, 1716, put an end to
this controversy, and Newton some time afterward published the
correspondence with the Abbé Conti, which had hitherto been only
privately circulated among the friends of the disputants.[71]
In 1722, a new edition of the Commercium Epistolicum was published,
and there was prefixed to it a general review of its contents, which
has been falsely ascribed to Newton.[72] When the third edition of
the Principia was published in 1725, the celebrated scholium which we
have already quoted, and in which Leibnitz’s differential calculus was
mentioned, was struck out either by Newton or by the editor. This step
was perhaps rash and ill-advised; but as the scholium had been adduced
by Leibnitz and others as a proof that Newton acknowledged him to be
an independent inventor of the calculus,—an interpretation which it
does not bear, and which Newton expressly states he never intended it
to bear,—he was justified in withdrawing a passage which had been so
erroneously interpreted, and so greatly misapplied.
In viewing this controversy, at the distance of more than a century,
when the passions of the individual combatants have been allayed,
and national jealousies extinguished, it is not difficult to form a
correct estimate of the conduct and claims of the two rival analysts.
By the unanimous verdict of all nations, it has been decided that
Newton invented fluxions at least ten years before Leibnitz. Some of
the letters of Newton which bore reference to this great discovery
were perused by the German mathematician; but there is no evidence
whatever that he borrowed his differential calculus from these letters.
Newton was therefore the _first_ inventor, and Leibnitz the _second_.
It was impossible that the former could have been a plagiarist; but it
was possible for the latter. Had the letters of Newton contained even
stronger indications than they do of the new calculus, no evidence
short of proof could have justified any allegation against Leibnitz’s
honour. The talents which he displayed in the improvement of the
calculus showed that he was capable of inventing it; and his character
stood sufficiently high to repel every suspicion of his integrity. But
if it would have been criminal to charge Leibnitz with plagiarism, what
must we think of those who dared to accuse Newton of borrowing his
fluxions from Leibnitz? This odious accusation was made by Leibnitz
himself, and by Bernouilli; and we have seen that the former repeated
it again and again, as if his own good name rested on the destruction
of that of his rival. It was this charge against Newton that gave
rise to the attack of Keill, and the publication of the Commercium
Epistolicum; and, notwithstanding this high provocation, the committee
of the Royal Society contented themselves with asserting Newton’s
priority, without retorting the charge of plagiarism upon his rival.
Although an attempt has been recently made to place the conduct of
Leibnitz on the same level with that of Newton, yet the circumstances
of the case will by no means justify such a comparison. The conduct
of Newton was at all times dignified and just. He knew his rights,
and he boldly claimed them. Conscious of his integrity, he spurned
with indignation the charge of plagiarism with which an ungenerous
rival had so insidiously loaded him; and if there was one step in his
frank and unhesitating procedure which posterity can blame it is his
omission, in the third edition of the Principia, of the references
to the differential calculus of Leibnitz. This omission, however,
was perfectly just. The scholium which he had left out was a mere
historical statement of the fact, that the German mathematician had
sent him a method which was the same as his own; and when he found
that this simple assertion had been held by Leibnitz and others as a
recognition of his independent claim to the invention, he was bound
either to omit it altogether, or to enter into explanations which might
have involved him in a new controversy.
The conduct of Leibnitz was not marked with the same noble lineaments.
That he was the aggressor is universally allowed. That he first dared
to breathe the charge of plagiarism against Newton, and that he often
referred to it, has been sufficiently apparent; and when arguments
failed him he had recourse to threats—declaring that he would publish
another Commercium Epistolicum, though he had no appropriate letters
to produce. All this is now matter of history; and we may find some
apology for it in his excited feelings, and in the insinuations which
were occasionally thrown out against the originality of his discovery;
but for other parts of his conduct we seek in vain for an excuse.
When he assailed the philosophy of Newton in his letters to the Abbé
Conti, he exhibited perhaps only the petty feelings of a rival; but
when he dared to calumniate that great man in his correspondence with
the Princess of Wales, by whom he was respected and beloved; when he
ventured to represent the Newtonian philosophy as physically false,
and as dangerous to religion; and when he founded these accusations on
passages in the Principia and the Optics glowing with all the fervour
of genuine piety, he cast a blot upon his name, which all his talents
as a philosopher, and all his virtues as a man, will never be able to
efface.