This document summarizes the development of calculus by Isaac Newton and Gottfried Leibniz in the late 17th century. It describes how Newton developed early concepts of calculus privately in the 1660s using notions of moments, fluents, and fluxions, which laid the foundations for differentiation and integration. However, Newton chose not to publish his work, while Leibniz independently developed and published similar calculus concepts in the 1670s and 1680s, sparking a bitter priority dispute between the two scientists. The document provides historical context on the precursors to calculus and the independent yet concurrent work of Newton and Leibniz in developing the fundamental principles and techniques of this field of mathematics.

1. Erik O. Mayer
The War for Calculus
Throughout history, the notion of war has typically connoted bloody affairs, violent
battles between countries, or coalitions in defense of, or in pursuit of, some nationalistic end.
Lives are lost, resources are consumed, much is sacrificed; but more often than not, the sacrifice
is deemed worthy of the cause. Unsurprisingly, the “war” that took place between Isaac Newton
and Gottfried Wilhelm Leibniz during the late 17th century and early 18th century over the
intellectual ownership of calculus bears little resemblance to an actual war. No blood was shed,
no lives were lost, no nations gained or lost their right to exist. No one side can even definitively
purport to have “won.” But despite those obvious dissimilarities, the consequences of this
calculus “war” were costly to the entire mathematical community, as any war is for its
participants. Discoveries that are heralded as among the most brilliant in the history of
mathematics are tainted by the ensuing demonstrations of petulance and pride, coloring an
unfortunate, ugly shade of humanity upon the portraits of two geniuses who loom large in the
history of mathematics.
Before delving into the specifics of the dispute that occurred between Newton and
Leibniz, it is important to note that it would be a gross oversimplification to proclaim that either
of these men “invented” calculus. Like any significant field in science or mathematics, calculus
developed over the course of decades and centuries, with many great mathematicians making
important contributions, dating as far back as Archimedes with his method of exhaustion to
determine quadratures (Boyer, 1959). As critical as Newton and Leibniz were to the
development of calculus, they certainly owed a great deal to their predecessors. Newton himself
once proclaimed, “If I have seen further, it is by standing upon the shoulders of giants” (BBC).

2. Calculus as we know it today addresses a vast assortment of problems, including
instantaneous velocities and accelerations, tangents to curves, extrema points of functions, and
areas beneath and between curves. Before Newton and Leibniz put forth their general analytical
methods, which could address all of these with ease, these problems were widely considered
distinct (Kline, 1972). Such problems were of great interest to many 17th century
mathematicians. Descartes worked on tangent lines and properties of curves, as did Fermat and
Roberval (Kline 1972). Kepler and Fermat researched and developed methods for determining
maxima and minima of specific kinds of curves (Kline, 1972). With regards to quadratures,
Cavalieri, Fermat, Pascal, Wallis, Torricelli, and Roberval all came upon the following result,
written here in our modern notation (Boyer, 1959):
The result is written in this manner out of convenience, but in doing so, we must be
careful not to give these mathematicians too much credit. They did not convey this result in a
generalized, methodological statement. Rather, this formula and results similar to it were
solutions to specific geometric problems (Boyer, 1959). And that word of caution must be
uttered in preface to most of these mathematicians’ results. It was exactly the generality and the
algebra of Newton and Leibniz’s approaches that made them a significant leap forward from
their predecessors.
There is an abundance of interesting work by these pre-Newton, pre-Leibniz
mathematicians in the 17th century, too much to describe in detail in the context of this paper.
The seeds of the important concepts were there for Newton and Leibniz to cultivate. Even the
inverse relationship between the tangent and area problems, the central tenet of the Fundamental

3. Theorem of Calculus, had been noticed in a geometric context by the likes of Fermat and Isaac
Barrow, Newton’s mentor (Boyer, 1959). Of course, contained within a narrow geometric
context, the importance of this theorem was not recognized. It would take a great mathematical
mind to pull together all these disparate threads into a cohesive methodology. The first of these
great minds was the Englishman Isaac Newton.
Upon arriving at Trinity College of Cambridge in 1661, Newton studied mathematics
substantially for the first time (Kline, 1972). By virtue of having him as his mentor, Newton
became familiar with Barrow’s work, and even helped Barrow prepare to publish Geometrical
Lectures (Boyer, 1959). Newton was also greatly influenced by the works of Descartes and John
Wallis, the latter of whom Newton credited with guiding his thinking towards his initial
breakthroughs (Boyer, 1959). In 1665, an outbreak of the plague in London forced Cambridge to
shut down, and subsequently, Newton returned to his home in Woolsthorpe. It was during this
isolated time at home, from 1665 to 1666, that Newton “built the foundations for his work in
optics, celestial mechanics, and mathematics, including the foundations of the calculus”
(Hellman, 2006).
Newton’s work on calculus evolved over time. He began with the notion of a “moment,”
an infinitely small increase of a variable, which he denoted by o. Similar work with
infinitesimals had been done previously by the likes of Fermat, James Gregory, and Wallis
(Boyer, 1959). Suppose the area under a curve, z, is given by (Kline, 1972):
z = axm .
Then by replacing x with x + o, and adjusting the area with the moment of the area, oy,
we get:
z + oy = a(x + o)m .

4. At this point, one of Newton’s great mathematical contributions reveals its usefulness.
Newton applies his binomial theorem to expand (x + o)m in an infinite series. Through further
algebraic manipulation, including dropping terms containing o, Newton obtains the result:
y = maxm – 1 .
What Newton has done here is determine the curve y that has an area z bounded beneath
it. And he did it by exploring the instantaneous rate of change of the area. In this example,
Newton has demonstrated the Fundamental Theorem of Calculus (Kline, 1972).
This process was adjusted with his notion of fluents and fluxions. If x and y are fluents,
the quantities generated, then ẋ and ẏ are the fluxions, the rates of generation of those fluents
(Boyer, 1959). The notion of the fluxion is quite similar to our modern-day conception of the
derivative. Newton’s process with fluxions proceeds quite similarly to his previous derivation.
He starts with the function y = x n, substitutes y + ẏo for y, and x + ẋo for x, uses the binomial
theorem to expand (x + ẋo)n, manipulates and cancels terms, then finally neglects terms still
containing o, as they “can be considered zero in comparison with the others” (Boyer, 1959). His
final result is ẏ = nx n – 1 ẋ, which is equivalent to what he derived previously.
Not entirely comfortable with the logic behind dropping infinitely small terms from his
calculations, Newton sought to further refine his methods with the notion of ultimate ratios
(Boyer, 1959). Results henceforth were described in the form of a ratio between the fluxions of
the abscissa and the ordinate. For example, in the same example from before, with y = x n,
Newton would state the result in the following manner: “the fluxion of x is to the fluxion of x n as
1 to nx n – 1 ” (Kline, 1972). In the process of this derivation, instead of simply dropping terms
containing the moment o, Newton allowed them to “vanish,” which hews closely to our modern-
day conception of the limit (Boyer, 1959).

5. With his fluents and fluxions, Newton had developed a general method for determining
tangents to curves, extrema points, and areas beneath and between curves. He united an
abundance of problems previously considered distinct under a single umbrella of techniques.
And yet, despite the power of his formulations, Newton decided against publishing them for the
entire scientific community to see. By 1671, Newton had compiled his work on calculus in
several different manuscripts, including De Analysi and Methodus Fluxionum, but shared them
with only a handful of British contemporaries (Katz, 2009). Newton’s choice not to publish
proved costly in the long run, but his reasons were not without merit.
Even if Newton had wanted to publish, circumstances in London at the time would have
made it difficult. In 1666, a devastating fire ravaged London and destroyed its publishing houses
(Bardi, 2006). In the wake of this great fire, a recession brought the book publishing industry to a
halt (Hellman, 2006). As a result, complex mathematical works in particular were a risky
proposition. For instance, a mathematical work by Isaac Barrow, Newton’s own mentor at
Cambridge, brought one publisher to the brink of bankruptcy (Bardi, 2006). Publishers had to be
extremely careful in their selection processes.
While this was certainly an obstacle for anyone trying to publish in London, in the face of
someone determined enough, it would not have been insurmountable. There were other avenues
toward publishing, particularly in the form of journals. Indeed, Newton himself did actually
publish a paper on optics in 1672, in Philosophical Transactions, the official journal of the Royal
Society, which is an English-based scientific society (Hellman, 2006). During the same period in
which Newton developed his calculus ideas, he also made great strides in the field of optics. In
his experiments, Newton determined that light was a particle, and that white light consisted of all
colors of the rainbow. These assertions were in direct contrast to the prevailing opinion of the

6. time, which asserted that light was a wave, and that white light was the absence of any color
(Bardi, 2006). Unfortunately for Newton, when he published his new ideas on light and color in
1672, the response from the optics community was not kind.
Chief among his critics was Robert Hooke, renowned British scientist, member of the
Royal Society, and proponent of the very ideas with which Newton’s paper disagreed. In
Newton’s paper, Hooke saw a threat to his intellectual prestige and did everything he could to
discredit Newton’s assertions. Appointed by the Society to lead a committee investigating the
validity of Newton’s claims, Hooke predictably ruled in his own favor (Bardi, 2006). Hooke
examined the experiments carried out by Newton, and concluded that Newton had interpreted the
results incorrectly. Against Hooke’s reputation in the scientific community, Newton could do
little to swing popular opinion in his favor (Bardi, 2006). Newton endured criticisms from Hooke
and other preeminent scientists for years following this publication. This experience
understandably discouraged Newton from publishing his work in the future (Kline, 1972).
Perhaps as a result of this suffocating criticism, Newton feared that his work on calculus
was not rigorous enough. Newton is said to have been “dissatisfied with the logical foundations
of the subject” (Boyer, 1959). Unlike the great Greek works in geometry, Newton’s calculus
lacked complete clarity and specificity in its definitions, and the proofs with which to defend its
methods (Katz, 2009). As already noted, Newton attempted to refine his methods to circumvent
the logically questionable act of dropping terms containing the moment. Mathematicians earlier
in the century, such as Cavalieri, were criticized for their use of indivisibles (Boyer, 1959).
Although Newton was supremely confident in the reliability of his methods to produce correct
results, without the rigor of proofs, he would be vulnerable to criticism.

7. And so Newton’s work was left unpublished, appreciated only by the fortunate
colleagues lucky enough to see his notes and manuscripts. The glory of his work’s significance
was left up for grabs. Had Newton proceeded differently, it seems likely that there would have
never been a dispute over priority with Leibniz, who, at the time of Newton’s most fertile
intellectual years in the mid-1660’s, barely had any deep knowledge of mathematics yet, let
alone the ingenius notions comprising the foundations of calculus that Newton possessed.
Born in the German city of Leipzig, Gottfried Wilhelm Leibniz would later attend the
University of Leipzig, studying philosophy and law (Bardi, 2006). Following the completion of
his university studies, Leibniz decided to pursue a career in law. However, in 1667, Leibniz
became acquainted with a German statesman named Boineburg, who put him on the path
towards diplomatic work. In 1672, the same year Newton published his controversial optics
paper, Leibniz was sent to Paris as an ambassador for the archbishop elector of Mainz (Bardi,
2006).
For the intellectually curious Leibniz, Paris provided an abundance of opportunity.
Shortly after arriving in Paris, Leibniz came across the Dutch physicist Christian Huygens, who
encouraged Leibniz to study mathematics deeply (Boyer, 1959). And so Leibniz did, consuming
the works of Wallis, Gregory St. Vincent, and Cavalieri, among other prominent 17th century
mathematicians. Starting in 1675, an extraordinarily short time after beginning his serious
mathematical studies, Leibniz began compiling his calculus methodology in manuscripts (Kline,
1972).
Later in life, Leibniz would give credit for his initial breakthroughs in calculus to the
differential triangle, which he happened upon in the work of Pascal (Boyer, 1959):

8. In this figure, Leibniz realized that the tangent line to the curve was determined by the
ratio of the differences in the ordinates and in the abscissas, that the area under the curve was
determined by the summation of infinitely thin rectangles, each with a height corresponding to
the ordinate, and that therefore, the two concepts had an inverse relationship (Boyer, 1959). So
clear was this relationship to Leibniz, he later remarked that “Pascal seemed to have a bandage
over his eyes” for not recognizing it (Boyer, 1959).
And so Leibniz’s calculus spawned from a close examination of sequences,
corresponding sequences of differences, and sums of differences. This quickly evolved into an
analysis of curves. The notation he developed is largely the same notation used in calculus today.
The quantity dx represents the difference in x-values, or abscissas, whereas dy represents the
difference in y-values, or ordinates. Leibniz used an elongated S, ∫, to represent a sum (Kline,
1972). He relied on the strength of his notation to produce many important results, including
what are now known as the power rules for differentiation and integration, the product rule, the
quotient rule (Boyer, 1959), as well as the chain rule, the formula for integration by parts, and
the formula for the volume of a solid of revolution about the x-axis (Kline, 1972). These results
were discovered by Leibniz during the mid-to-late 1670’s, about a decade after Newton came
upon many of the same results, albeit in a different notational form.
Although Leibniz’s work rested on the same questionable logic that Newton’s did, in that
there was no formal conception of the limit, many of the definitions were not clear, and most of

9. the results lacked formal proofs, Leibniz’s confidence in the symbolic utility of his work
outweighed the potential for criticism. And thus, in 1684, he published in Acta Eruditorum, a
German journal he cofounded, a six-page paper whose title translates into English as “A New
Method for Maxima and Minima as Well as Tangents, Which is Neither Impeded by Fractional
nor Irrational Quantities, and a Remarkable Type of Calculus for Them” (Bardi, 2006). With
this, Leibniz had beaten Newton and became the first person to publish a work on differential
calculus.
Thus far, the stories of Newton and Leibniz have been described separately. Had their
stories truly been separate, Newton and his supporters would have had no choice but to
acknowledge that Leibniz’s developments were just as legitimate as Newton’s. As fate would
have it, though, their intellectual paths did cross in the 1670’s, and this fact would be used as
ammunition against Leibniz later, fairly or not.
In 1673, a year after he moved to Paris and began his serious study of mathematics,
Leibniz was invited to demonstrate his prototype for a mechanical calculator to the Royal
Society in London. Perhaps unsurprisingly, he was criticized by Robert Hooke (Hall, 1980).
During his visit, in an attempt to impress his company, Leibniz shared a discovery he thought he
had made about the differences between terms in certain series with the prominent
mathematician Pell. Pell responded that the result Leibniz was describing had already been
discovered and published by another mathematician (Hall, 1980). Embarrassed, Leibniz wrote an
account of how he came about the result himself to preempt any suspicion that he may have
plagiarized (Bardi, 2006). Leibniz also learned that a discovery he had made about the circle
series had already been attained by British mathematicians Newton and Gregory (Hall, 1980).
These two revelations highlighted Leibniz’s inexperience and piqued his interest in what British

10. mathematicians had already been able to accomplish. During his visit, Leibniz studied several
published works by the British, including that of Barrow (Hall, 1980).
More important, however, were the contacts Leibniz developed in London. Impressed
with Leibniz’s knowledge and ability, Henry Oldenburg, the German-born secretary of the Royal
Society, kept Leibniz informed on British developments (Bardi, 2006). In cooperation with the
older British mathematician John Collins, Oldenburg sent Leibniz reports on mathematical
results in England over the next few years. Collins was well-connected with many British
mathematicians, including Newton (Hall, 1980). And so, it was through Leibniz’s
correspondence with Oldenburg and Collins that Leibniz became familiar with Newton.
The culmination of Leibniz’s correspondence with Oldenburg was a brief correspondence
directly with Newton in 1676. Leibniz was still curious about Newton’s work on series, and also
had heard rumors that Newton had his own calculus method (Hall, 1980). It is vitally important
to note that at this point, Leibniz already had the central pieces of his own method in place.
Newton sent two letters to Leibniz, both very complimentary and cordial. In the first letter,
Newton made no reference to his calculus methods, discussing only the binomial theorem and
some of his results in series (Hall, 1980). Despite the letter containing very little, if any, new
information for Leibniz, he was appreciative, and he responded enthusiastically the very next
day. However, the date of dispatch for Newton’s letter was mistakenly marked as three weeks
earlier than it was actually posted. This would later create the impression that Leibniz had
thoroughly studied the letter before responding (Hall, 1980).
Newton sent his second letter only a few months later. Again, this letter was quite
friendly, but Newton was still hesitant to reveal his proudest mathematical discoveries to
someone he viewed as inexperienced in comparison to him. And so, Newton did make reference

11. to his method of fluents and fluxions, but he concealed it in the form of an anagram, making it
impossible for Leibniz to decipher (Katz, 2009). In 1676, Leibniz’s tenure as ambassador in
Paris came to an end, and a new job as librarian to the Duke of Hanover awaited him back in
Germany. Due to this move, it took Newton’s second letter several months to reach Leibniz
(Hall, 1980). This delay in Leibniz’s acquisition of the second letter again created the impression
that Leibniz carefully studied Newton’s words before responding. In reality, Leibniz responded
eagerly and quickly, even going so far as to discuss some of his own work on tangent lines (Hall,
1980). Newton did not respond to this letter, which arrived close to a year after he had sent his
letter to Leibniz.
While Leibniz was in transit from Paris to Hanover, he made his second visit to London.
During this visit, Leibniz finally met John Collins. Collins, the man with a vast amount of
connections to mathematicians in England, possessed copies of several of Newton’s papers,
including De Analysi (Hall, 1980). It was at this point that Leibniz had the opportunity to look
through these papers (Hellman, 2006). How much information Leibniz was able to glean from
these papers is debatable, but considering this visit occurred after Leibniz had conceived the
central tenets of his calculus method, there would not have been much new for him in relation to
that. Regardless, later accusations of plagiarism hinged on this moment.
After Leibniz’s second visit to London, and after his exchange of letters with Newton in
1676, communication between Leibniz and anyone in London ceased. This is explained
primarily by the subsequent deaths of Oldenburg and Collins, in 1677 and 1683 respectively
(Hall, 1980). With Oldenburg gone, Leibniz no longer had a direct contact in London.
Consequently, matters remained quiet until Leibniz published his calculus work in 1684.

12. By 1684, Newton’s attention had shifted away from mathematics and to his seminal work
on mechanics and physics, which he was in the process of compiling and preparing for
publication as the Principia (Hall, 1980). It seemed as if the only times he would speak on
mathematics in particular was in assertion of his priority. David Gregory, nephew of esteemed
British mathematician James Gregory, published in 1684 a work on series, based largely off of
his uncle’s notes. This publication contained several results that Newton had also attained, but
had kept unpublished (Hall, 1980). David Gregory was deferential in the matter, sending Newton
a copy of his manuscript to peruse. Unconcerned, Newton did not take any drastic action, such as
rush to publish his own work, or demand that Gregory yield. He simply made note of the fact
that he had attained these same results earlier, by later making public the letters he sent to
Leibniz (Hall, 1980).
Perhaps it was this experience with David Gregory that colored Newton’s early response
to Leibniz’s publication. Perhaps he thought he could easily make note of his priority, as he did
in the Gregory situation. Perhaps he underestimated Leibniz’s methods in comparison to his
own, thinking that his would prevail once they were known at large. Whatever the reason,
Newton remained civil and complimentary for a number of years. The conflict simmered slowly,
growing more potent and ready to burst as time went on.
Newton’s quiet assertion of his ownership of calculus came in the form of a brief
description of his methods in comparison to Leibniz’s in his Principia, which he finally
published in 1687 (Hall, 1980). The reference to Leibniz is not in the least bit hostile; it is, rather,
quite deferential. Newton says that Leibniz “had fallen on a method of the same kind, and
communicated to me this method, which scarcely differed from mine, except in notation and the

13. idea of the generation of quantities” (Hellman, 2006). No indication yet existed of the future
animosity that both men would harbor for each other.
Interestingly, Newton’s description of fluxions in Principia was dense and not explained
very well (Hall, 1980). It seemed more than anything like Newton’s attempt to stake his claim on
calculus. Moreover, besides this brief, self-contained mention of fluxions, the mathematics
contained in Principia was cast entirely in a classical geometric context (Bardi, 2006). Newton’s
aforementioned uncertainty over the logical foundations of his calculus surely influenced his
decision to employ mathematically accepted geometric methods and proofs in Principia.
In the early 1690’s, Wallis, who was, like Newton, a British mathematician, implored
Newton to allow him to publish some of Newton’s calculus in a collection of British
mathematical works (Hellman, 2006). Newton obliged, and Wallis reproduced and expanded
upon Newton’s two letters to Leibniz in his Works, giving the method of fluxions a more proper
explanation than Newton had done in his own Principia. With this, Wallis created the impression
that a proper explanation had actually been included in Newton’s letters to Leibniz, when in
reality, Newton’s only reference to his fluxions had been contained in the anagram (Hall, 1980).
Perhaps in an attempt to support his fellow Englishman, Wallis also asserted that Newton’s
fluents and fluxions were easier to use than Leibniz’s method (Bardi, 2006).
If these remarks stirred any anger in Leibniz, it was not directed at Newton. The two men
exchanged more complimentary letters in 1693, where Leibniz praised Newton and encouraged
him to continue on with mathematical work. Newton responded by calling Leibniz one of the top
mathematicians in the world (Bardi, 2006). Whether Newton’s attitude toward Leibniz was
genuine or merely polite, it certainly seems as if Leibniz respected the former a considerable
amount. He was sorry to hear that Newton had started working at the British mint in 1696,

14. further removing him from serious intellectual pursuits (Bardi, 2006). Following the publication
of his calculus in 1684, he had amassed a great deal of renown across the European continent, so
perhaps he underestimated the threat that Newton posed. Some of Leibniz’s supporters, however,
did not.
Esteemed Swiss mathematicians and brothers Johann and Jakob Bernoulli encountered
Leibniz’s calculus after it was published in 1684, and, sensing its significance, they immediately
began championing it and Leibniz (Hellman, 2006). They worked directly with Leibniz to help
him better express his ideas and demonstrate their usefulness. It was due in part to the Bernoullis
that Leibniz’s calculus spread across the continent as prodigiously as it did, encouraging more
mathematicians like L’Hôpital to use it (Bardi, 2006). After hearing that Newton had developed
his own calculus method, separate from Leibniz’s, Johann Bernoulli was skeptical, and instigated
the first major battle in the so-called calculus “war” in 1696.
This battle between mathematicians, of course, came in the form of a mathematical
challenge. Issued as a problem in the Acta Eruditorum to all of the eminent mathematicians of
the day, Bernoulli’s challenge was in reality conceived to test the formidability of Newton’s
calculus (Hellman, 2006). The problem, known as the brachistochrone problem, tasked all
attempters with finding the curve connecting two non-vertically aligned points along which a
body would descend the quickest. In essence, this is an optimization problem, one that would be
difficult to solve without calculus. The solution, which is part of a cycloid, was successfully
attained by the likes of Johann and Jakob Bernoulli, L’Hôpital, Leibniz, and, indeed, Newton,
who is said to have solved the problem on the day he received it, after a long day working at the
mint (Bardi, 2006). Newton sent his solution anonymously to the Royal Society, but Johann

15. quickly deduced its author in the same manner one could detect a “lion from his claw” (Hellman,
2006). With this, Newton had proved his methods were formidable.
To reiterate, however, Leibniz himself did not feel threatened by Newton at this point.
Despite the few references to Newton’s methods now in print, Leibniz was widely heralded as
the “inventor” of calculus (Bardi, 2006). In some respects, he fancied himself as the leader of
some exclusive calculus club, of which Newton was a part. In celebration of Johann’s challenge,
Leibniz wrote a review of the submitted solutions in the Acta Eruditorum, touting them as
demonstrations of the calculus he created (Hellman, 2006). Although Leibniz did not directly
claim that Newton was using his methods, the perception and the implication, perhaps
unintended, was there.
Such an implication, unintended or not, was enough for one of Newton’s supporters to
lash out. Nicolas Fatio de Duillier, a Swiss mathematician, became personally close with Newton
after relocating to England in the late 1680’s (Hall, 1980). For reasons not entirely clear, their
relationship and correspondence effectively ended after 1693. Despite this, Fatio remained fond
of Newton, and sided with him over the question of calculus priority. Interestingly, Fatio had
studied under Huygens, like Leibniz had done, and as a result, he likely considered himself a
peer of Leibniz’s, despite his dramatically lesser abilities and younger age. Leibniz was not
particularly interested in Fatio’s attempts at correspondence with him, causing Fatio to feel
slighted. Furthermore, Fatio himself had also solved the brachistochrone problem and submitted
a solution, but his was ignored (Bardi, 2006). Fatio’s loyal feelings towards Newton and his
resentful feelings towards Leibniz caused him to write his own analysis of the brachistochrone
problem in 1699, in which he criticized Leibniz heavily.

16. In his criticism, Fatio declared that Newton was the inventor of calculus, and mentioned
the possibility that Leibniz had borrowed from Newton. Furthermore, he mocked Leibniz’s
repeated claims of ownership of calculus as vain, while simultaneously praising Newton’s
silence in the matter as modest (Hellman, 2006). Understandably, Leibniz was outraged, and
replied directly with a defense published in Acta Eruditorum. In his defense, Leibniz cited his
cordial relationship with Newton and the respect they had for each other, implying that Newton
would certainly not support Fatio’s claims (Bardi, 2006). Sure enough, Newton remained
completely silent on the matter, which seemed to confirm Leibniz’s assertions. On top of that,
Leibniz appealed to the Royal Society for support, and its members readily sided with him over
Fatio (Bardi, 2006). Fatio was largely ignored in light of this, seen merely a bitter mathematician
lashing out at someone of much greater talent and achievement. Unfortunately, if this experience
convinced Leibniz that the Royal Society would always be a reliable ally, he was gravely
mistaken.
For Newton, whose health had diminished slightly in the last years of the 17th century,
and whose career had shifted away from intellectual pursuits, an opportunity to step back into the
spotlight presented itself when his nemesis Hooke died in 1703 (Hellman, 2006). At long last,
Newton could share his work freely, knowing that his most vicious critic was out of the picture.
The same year, Newton was elected the president of the Royal Society, bringing him tremendous
prestige (Hellman, 2006). And thus, in 1704, Newton finally published his major work on optics,
appropriately titled Opticks. As an appendix to this work, Newton included a couple of papers he
wrote on his calculus (Bardi, 2006). It was in response to this that Leibniz made his most
damaging error.

17. To some extent, Fatio was right about Leibniz. His vanity, his need to regularly promote
himself, would be his downfall. After Newton published Opticks, Leibniz reviewed Opticks in
the Acta Eruditorum, most crucially the appendixes on calculus (Bardi, 2006). In his review,
Leibniz stated that Newton’s use of fluxions was similar to his own methods, just contextualized
in terms of motion. This much was true. But Leibniz then compared this to Cavalieri’s use of
indivisibles and Honoré Fabri’s recasting of Cavalieri’s indivisibles in a motion-based context
(Hall, 1980). While Leibniz probably thought the motion analogy was apt, Fabri quite blatantly
borrowed Cavalieri’s ideas and merely presented them in a slightly different fashion. In making
such a comparison, Leibniz insinuated, intentionally or not, that his ideas came first and that
Newton had subsequently reworked them into a motion context (Hall, 1980). Once he became
aware of this comparison, Newton would no longer remain silent.
In 1708, Scottish mathematician John Keill published a paper in the Royal Society’s
Philosophical Transactions, in which he blatantly accused Leibniz of plagiarizing Newton
(Hellman, 2006). When Leibniz became aware of this attack, he likely thought he could refute it
in the same way he had done with Fatio’s attack, by appealing directly to the Royal Society and
Newton. Leibniz was wrong.
By this time, Newton had been president of the Royal Society for several years,
dramatically increasing his popularity and influence. In appealing to the Royal Society, Leibniz
was essentially appealing directly to Newton, putting his reputation at Newton’s mercy. And it
was at this crucial juncture that Keill showed to Newton the unfortunate comparison Leibniz
made in his review of Opticks (Hall, 1980). When Leibniz demanded an apology from Keill,
Keill refused to acquiesce, and instead repeated his claims with even more force to the Royal
Society, citing Leibniz’s Opticks review as proof that he was not attacking Leibniz unfairly

18. (Hall, 1980). Still wanting to bury the accusations of plagiarism, Leibniz demanded an official
redress from the Royal Society.
The Royal Society, guided by Newton, proceeded to investigate the claims. The results of
this investigation were published in a report named the Commercium Epistolicum in 1713 (Hall,
1980). In the Commercium, Leibniz was depicted as someone with a history of plagiarizing,
citing the incident with Pell decades earlier. Leibniz’s correspondence with Oldenburg and
Collins was mined for suspicious excerpts, and Collins’ possession of Newton’s unpublished
calculus papers proved, at least in the Society’s view, that Leibniz carefully studied Newton’s
work. And of course, Newton’s letters with Leibniz in 1676, whose actual content was already
distorted by Wallis’s Works, painted Leibniz as mathematically inexperienced and suspiciously
curious of Newton’s results (Bardi, 2006). Newton, who had remained quiet on this matter for
decades, suddenly dealt a devastating blow to Leibniz with this report.
This was the point of no return, the moment at which the slowly escalating conflict boiled
over. The façade of respect and cordiality between these two men collapsed. Leibniz tried to
defend himself, tried to accuse Newton of being the plagiarist. In the Charta Volans, Leibniz
wrote that the only material on calculus Newton had published up until that point, in Principia,
Wallis’s Works, and elsewhere, could have very easily been reformulations of ideas that Leibniz
already published. He accused the British of being xenophobic, of trying to steal all of the glory
of inventing calculus for themselves (Bardi, 2006). Johann Bernoulli made note of an error in the
first edition of Principia, in which Newton failed to accurately acquire a second differential
(Hall, 1980). Newton made further attacks on Leibniz in his Account of the Commercium
Epistolicum, calling into question Leibniz’s mathematical ability at the time of their 1676 letters,
reasoning that he could not have conceived his calculus until at least 1677. He also touted his

19. own method of fluxions as superior to Leibniz’s (Hall, 1980). In subsequent editions of
Principia, Newton removed the acknowledgment of Leibniz entirely (Hellman, 2006).
The dispute continued in this manner until Leibniz’s death in 1716, and beyond,
becoming more and more personal with each barb exchanged. Leibniz, having become desperate,
tried to reframe the conflict by attacking Newton’s theory of gravitational attraction and his
general philosophy (Bardi, 2006). But it was too late for Leibniz. His credibility was irreparably
damaged the moment the Royal Society issued the Commercium. And there was no chance for
reconciliation. Even after Leibniz’s death, Newton continued to bolster his case, publishing an
even more scathing second edition of the Commercium in 1722 (Hall, 1980). Newton did not
stop advocating for himself at the expense of Leibniz until he died in 1727 (Hall, 1980).
After the deaths of both Newton and Leibniz, the residual effects of this “war” could be
felt for decades thereafter. It may seem that Newton prevailed over Leibniz in terms of
perception, but Newton’s mathematical victory was not decisive in the least. While Newton’s
British supporters, who worshipped him with an intense fervor, continued to employ fluxions
long after Newton was dead, the European continental mathematicians, led by the Bernoullis,
used Leibniz’s notation to make significantly greater advances in the 18th and 19th centuries
(Hellman, 2006). British mathematicians, who remained passionately loyal to Newton, ignored
these advances made possible by Leibniz’s formulations and consequently fell behind (Bardi,
2006). This schism in the mathematical community stubbornly persisted well into the 19th
century, until the calculus of Leibniz slowly became the standard.
Both Newton and Leibniz demonstrated mathematical genius in independently
developing calculus as we know it today. But like all men, these geniuses had human flaws,
insecurities that ultimately betrayed deeper passions and egotism. And consequently, one of the

20. greatest intellectual achievements in history is inextricably linked with a war of petulance whose
size can only be matched by the enormity of the achievement in question.
References
Bardi, Jason Socrates. The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical
Clash of All Time. New York: Basic Books, 2006. Print.
Boyer, Carl. The History of the Calculus and its Conceptual Development. New York: Dover
Publications, 1959. Print.
Edwards, Charles Henry Jr. The Historical Development of the Calculus. New York: Springer-
Verlag, 1979. Print.
Feingold, Mordechai. “Newton, Leibniz, and Barrow Too: An Attempt at a Reinterpretation.”
Isis 84.2 (1993): 310-338.
Hall, Alfred Rupert. Philosophers at War: The Quarrel Between Newton and Leibniz.
Cambridge: Cambridge University Press, 1980. Print.
Hellman, Hal. Great Feuds in Mathematics. Hoboken: John Wiley & Sons, Inc., 2006. Print.
Katz, Victor J. A History of Mathematics: An Introduction, 3rd Edition. Boston: Addison-Wesley,
2009. Print.
Kline, Morris. Mathematical Thought from Ancient to Modern Times Volume 1. New York:
Oxford University Press, 1972. Print.
“Sir Isaac Newton.” BBC Learning English. BBC, 2009.
http://www.bbc.co.uk/worldservice/learningenglish/movingwords/shortlist/newton.shtml