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Newton and Leibniz: The Story of Calculus

Two inventors, two notations, one calculus.

In the 1660s and 1670s Isaac Newton in England and Gottfried Wilhelm Leibniz in Germany each found the same powerful method for slopes and areas, independently. Newton found it first; Leibniz published first. The quarrel that followed split European mathematics for a century, and the notation you use today is a result of it.

On this page: a tap-through timeline from Archimedes to Weierstrass, the two notations side by side with one example worked in both, how the dispute shaped British and Continental mathematics, six primary-source excerpts, nine discussion and practice questions, and an eight-question quiz.

Tap through the timeline Compare the notations

One derivative, two notations. The slope of y = x³ at x = 2 is 12 whether you write it with Newton’s dots or Leibniz’s d’s.

Section IA timeline of calculus

Calculus was not invented in a single moment. Ideas about areas go back to ancient Greece, tangents to the 1600s, and a fully rigorous foundation to the 1800s. Newton and Leibniz sit in the middle, turning many separate tricks into one method.

Tap a milestone, or use the Previous and Next buttons (or the arrow keys on a selected row), to read about it. Rows are evenly spaced, not to scale: the gap from Archimedes to Fermat alone is about 1,860 years.

c. 225 BC: Archimedes squares the parabola

Archimedes (287–212 BC) showed that a segment of a parabola has area 4/3 of the triangle with the same base and vertex. He filled the segment with ever smaller triangles, whose areas form the series A(1 + 1/4 + 1/16 + …), and used the method of exhaustion to make the argument rigorous. His methods anticipated the integral calculus by nearly 2,000 years.

The tap-through timeline needs JavaScript turned on. Here are all eleven milestones.

c. 225 BC: Archimedes squares the parabola. Archimedes (287–212 BC) showed that a segment of a parabola has area 4/3 of the triangle with the same base and vertex. He filled the segment with ever smaller triangles, whose areas form the series A(1 + 1/4 + 1/16 + …), and used the method of exhaustion to make the argument rigorous. His methods anticipated the integral calculus by nearly 2,000 years.
1630s: Fermat: maxima, minima and tangents. Pierre de Fermat replied to a letter from Marin Mersenne on 26 April 1636 and went on to send the Paris mathematicians his Method for determining Maxima and Minima and Tangents to Curved Lines. His tangent method was one of the most important steps before calculus itself.
1665–66: Newton’s plague years. Plague closed Cambridge University in the summer of 1665, and Isaac Newton went home to Lincolnshire. In less than two years he made revolutionary advances in mathematics, optics and physics. In October 1666 he wrote a tract on fluxions; it was not published, but it was seen by many mathematicians.
1670: Barrow’s Lectiones Geometricae. Isaac Barrow, Newton’s teacher at Cambridge and the first Lucasian Professor of Mathematics, published his geometry lectures in 1670. They include a method of tangents using what is now called Barrow’s differential triangle, and Barrow is credited with recognising that differentiation and integration are inverse operations.
1675: Leibniz writes ∫ for the first time. Gottfried Wilhelm Leibniz, working in Paris, used the integral sign in a manuscript of 21 November 1675, writing ∫ f(x) dx. The long ∫ is a stretched S, for summa, a sum. The same manuscript gives the product rule for differentiation.
1684: Nova Methodus is published. Leibniz published his differential calculus in the journal Acta Eruditorum in 1684, as “A new method for maxima and minima, and for tangents…”. It introduced the d notation and the rules for powers, products and quotients, but gave no proofs. Jacob Bernoulli called it an enigma rather than an explanation. In 1686 Leibniz’s ∫ first appeared in print.
1687: Newton’s Principia. Newton’s Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) appeared in 1687, setting out his laws of motion and universal gravitation. His calculus itself was mostly unpublished: the Method of Fluxions, written in 1671, was not printed until 1736.
1712: The Royal Society rules for Newton. In a letter printed in the Royal Society’s Philosophical Transactions for 1708, John Keill accused Leibniz of plagiarism; Leibniz read it in 1711. Leibniz appealed to the Society, of which Newton was President. A committee appointed on 6 March 1712 reported on 24 April 1712 in Newton’s favour, without asking for Leibniz’s side. Newton himself drafted the report (the copy handed in was in Edmond Halley’s handwriting), which was printed as Commercium epistolicum and circulated early in 1713.
1734: Berkeley’s The Analyst. Bishop George Berkeley attacked the foundations of calculus in The Analyst (1734). Fluxions and infinitely small quantities, he argued, were treated as something when dividing and as nothing when they were dropped. Colin Maclaurin replied with an attempt to put Newton’s calculus on a rigorous footing.
1821: Cauchy’s Cours d’analyse. Augustin-Louis Cauchy’s Cours d’analyse (1821), written for students at the École Polytechnique in Paris, developed the basic theorems of calculus as rigorously as possible, building on limits. He also gave a rigorous definition of the integral.
1859–60: Weierstrass and the foundations. Karl Weierstrass, teaching in Berlin, tackled the foundations of analysis for the first time in his lectures “Introduction to analysis” in 1859–60, covering continuity and differentiability with new care. His approach to rigour shaped the precise style of analysis taught today.

Section IITwo notations, one idea

Newton thought of quantities as flowing in time. A flowing quantity x is a fluent, and its speed of flow is its fluxion, written with a dot: ẋ. Leibniz thought of a curve as made of infinitely many tiny pieces. A tiny difference in x is dx, and a sum of infinitely many pieces is ∫, a long S for summa.

Newton: fluxions

Rate of change: ẏ/ẋ
Second fluxion: ÿ
Integral: find the fluent whose fluxion is given
Time is the hidden variable

Example, y = x³. Let x grow by a tiny moment ẋo while y grows by ẏo:

y + ẏo = (x + ẋo)³Substitute= x³ + 3x²ẋo + 3xẋ²o² + ẋ³o³Expandẏ = 3x²ẋ + 3xẋ²o + ẋ³o²Subtract y = x³, divide by oẏ = 3x²ẋ, so ẏ/ẋ = 3x²Let o vanish

Backwards: the fluent of 3x²ẋ is x³.

Leibniz: differentials

Rate of change: dy/dx
Second derivative: d²y/dx²
Integral: ∫ f(x) dx, a sum of strips
The variable is written in the symbol

Example, y = x³. Let x grow by an infinitely small dx:

dy = (x + dx)³ − x³Difference in y= 3x²dx + 3x(dx)² + (dx)³Expanddy = 3x²dxDrop higher-order termsdy/dx = 3x²Divide by dx

Backwards: ∫ 3x² dx = x³ + C.

Both give a slope of 3 · 2² = 12 at x = 2. The mathematics is identical; only the symbols differ. Leibniz’s symbols survived in most of mathematics because they show which variable you are differentiating with respect to and make rules like the chain rule, dy/dx = (dy/du)(du/dx), look natural. Newton’s dots survive in physics for rates with respect to time.

Section IIIThe priority dispute

Newton developed his method of fluxions in 1665–66 but did not publish it; his October 1666 tract and later papers circulated among a few mathematicians. Leibniz developed his calculus in Paris in 1675–76 and published it in 1684 and 1686. Leibniz had visited London and exchanged letters with Newton through Henry Oldenburg, the Royal Society’s secretary, so Newton’s supporters came to suspect he had seen Newton’s ideas.

In a letter printed in the Royal Society’s Philosophical Transactions for 1708, John Keill accused Leibniz of plagiarism. Leibniz read it in 1711 and appealed to the Society, but its President was Newton. The committee it set up in 1712 did not ask Leibniz for his account, and Newton drafted its report, which decided for Newton. The report was printed as Commercium epistolicum early in 1713. Leibniz learnt of its contents that year from a letter by Johann Bernoulli and replied with an anonymous pamphlet, the Charta volans, dated 29 July 1713; he did not see a printed copy of the report until the autumn of 1714. He died in 1716 with the matter unresolved.

Today Leibniz’s discovery is regarded as independent of Newton’s. Newton was first to the method; Leibniz was first to publish, and his notation was better.

Section IVHow the dispute shaped British and Continental mathematics

On the Continent, Leibniz’s notation spread quickly. Jacob and Johann Bernoulli took up his calculus and developed it further, and Continental calculus grew up in the language of d and ∫.

In Britain, loyalty to Newton meant fluxions and dots. Cambridge continued to teach Newton’s calculus without reference to Leibniz’s methods, which made the newest Continental work harder for British students to read and to build on.

The return. In 1812 Charles Babbage and friends founded the Analytical Society at Cambridge to bring modern Continental mathematics there. Babbage, John Herschel and George Peacock published an English translation of Lacroix’s calculus textbook in 1816 and a book of examples in 1820. Babbage recalled working with equal facility in all three notations, “the dots of Newton, the d’s of Leibniz, or the dashes of Lagrange”; in the end the d’s won.

The foundations. Berkeley’s 1734 criticism showed that neither fluxions nor infinitely small differences had a logical basis. The answer came from the Continent: Cauchy’s Cours d’analyse (1821) built calculus on limits, and Weierstrass’s Berlin lectures of 1859–60 gave its foundations the precision taught today.

Section VSix primary sources

Each excerpt is marked as a quotation or a paraphrase, with where it comes from. Paraphrases restate a source in modern English; they are not the author’s words.

Every segment bounded by a parabola and a chord is equal to four-thirds of the triangle which has the same base and the same vertex.
Paraphrase, not a direct quotation: Archimedes, Quadrature of the Parabola (3rd century BC). Paraphrase, following MacTutor.

“…in November had the direct method of fluxions [i.e., the differential calculus] & the next year in January had the Theory of colours & in May following I had entrance into [th]e inverse method of fluxions [i.e., the integral calculus]. … All this was in the two plague years of 1665 & 1666. For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more then [sic] at any time since.”
Quotation: Isaac Newton, private memorandum written decades later, recalling 1665–66. Quoted, with the editor’s bracketed notes, in J. Stachel, introduction to Einstein’s Miraculous Year (Princeton University Press, 2005).

Here, for the first time, the area under a curve is written as ∫ f(x) dx, a sum of infinitely many thin strips, and the product rule for differences is set down.
Paraphrase, not a direct quotation: Gottfried Wilhelm Leibniz, manuscript of 21 November 1675. Paraphrase of its content, following MacTutor.

“That the differential method is one and the same with the method of fluxions, excepting the name and the mode of notation…”
Quotation: Report of the Royal Society committee, 24 April 1712, point IV, in the English given by David Brewster, Memoirs of Sir Isaac Newton, vol. 2 (1855), via The Newton Project.

“They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?”
Quotation: George Berkeley, The Analyst (1734), on Newton’s fluxions and vanishing increments. Quoted in MacTutor.

“…periodical meetings for the propagation of d’s; and consigned to perdition all who supported the heresy of dots.”
Quotation: Charles Babbage, Passages from the Life of a Philosopher (1864), on the Analytical Society he helped found at Cambridge in 1812. Quoted in MacTutor.

Section VIDiscussion and practice

Think or write an answer first, then open the reveal to compare.

Use Newton’s method of moments to find the fluxion of y = x². Replace x by x + ẋo and y by y + ẏo. Notation
Show answer and discussion
y + ẏo = (x + ẋo)² = x² + 2xẋo + ẋ²o²Substituteẏo = 2xẋo + ẋ²o²Subtract y = x²ẏ = 2xẋ + ẋ²oDivide by oẏ = 2xẋ, so ẏ/ẋ = 2xLet o vanish
✓ In Leibniz’s notation this is dy/dx = 2x, the same answer.
Translate Newton’s ẏ = 4x³ẋ into Leibniz’s notation, and say what function y could be. Notation
Show answer and discussion
dy/dx = 4x³Divide by ẋ; ẏ/ẋ becomes dy/dxy = x⁴ + CAn antiderivative (Newton: the fluent)
✓ Check: d(x⁴)/dx = 4x³.
The parabola y = x² and the chord y = 1 bound a segment. Find the area of the triangle with vertices (−1, 1), (1, 1), (0, 0), then use Archimedes’ 4/3 rule to predict the segment’s area. Check with an integral. Archimedes
Show answer and discussion
triangle = ½ · 2 · 1 = 1Base 2, height 1segment = 4/3 · 1 = 4/3Archimedes∫₋₁¹ (1 − x²) dx = 2 − 2/3 = 4/3Calculus agrees
✓ The series 1 + 1/4 + 1/16 + … also sums to 1/(1 − 1/4) = 4/3.
How many years passed between Newton’s October 1666 tract and Leibniz’s 1684 paper? And between the 1684 paper and the 1712 committee report? Dates
Show answer and discussion
1684 − 1666 = 18 yearsTract to first publication1712 − 1684 = 28 yearsPublication to report
✓ Newton had the method first; Leibniz published first. That gap is the heart of the dispute.
Compute ((x + o)² − x²)/o for o ≠ 0, then explain Berkeley’s objection to the next step, “let o be nothing.” Berkeley
Show answer and discussion
((x + o)² − x²)/o = (2xo + o²)/o = 2x + oValid only because o ≠ 0o = 0 gives 2xBut now o is zero after dividing by it
✓ Berkeley’s point: o cannot be non-zero for the division and zero afterwards. Cauchy and Weierstrass replaced “o vanishes” with a limit: 2x + h → 2x as h → 0, with h never equal to 0.
Leibniz writes the chain rule as dy/dx = (dy/du)(du/dx). Why might this notation have helped Continental mathematicians? Discussion
Show answer and discussion
The notation looks like fractions cancelling, so the rule is easy to remember and to apply, even though dy/du is not truly a fraction. Leibniz chose symbols that do some of the thinking for you. Newton’s dots are compact but hide which variable you are differentiating with respect to: ẏ means a rate with respect to time.
✓ There is no single right answer; good answers mention memorability, the variable shown in the denominator, and ease with substitution.
The committee did not ask Leibniz for his account, and Newton drafted its report. What does that suggest about how far to trust the 1712 verdict? Discussion
Show answer and discussion
A fair inquiry hears both sides and is not judged by one of the parties. Here the Royal Society’s President, Newton, drafted the report himself, so it is evidence of Newton’s view, not an independent verdict. Leibniz’s discovery is now regarded as independent: Newton first, Leibniz first in print.
✓ Good answers separate “who was first” from “who copied”: being first does not prove the other copied.
Why might Cambridge students before 1812 have found the newest Continental papers hard to read? Discussion
Show answer and discussion
Continental work, from the Bernoullis onward, was written with Leibniz’s d and ∫, while Cambridge taught Newton’s fluxions and dots. Loyalty to Newton after the dispute kept the British largely apart from the new methods until Babbage, Herschel and Peacock’s Analytical Society pushed to bring them in.
✓ Mention notation, national loyalty after the dispute, and the 1816 Lacroix translation.
Newton writes a second fluxion with two dots, ẍ. If a position is x = t³ at time t, find ẋ and ẍ, and write both in Leibniz’s notation. Notation
Show answer and discussion
ẋ = 3t², i.e. dx/dt = 3t²Velocityẍ = 6t, i.e. d²x/dt² = 6tAcceleration
✓ Dot notation is still used in physics for time derivatives, so both systems survive.

Section VIISelf-guided quiz: eight quick questions

Choose one answer for each question, then press “Check my answers.” Your best score is saved only in this browser.

1. What did Newton call the rate of change of a flowing quantity?

A differentialA fluentA fluxionAn integral

2. In which year did Leibniz first publish his differential calculus, in Acta Eruditorum?

1666168416871712

3. Who drafted the 1712 Royal Society committee report on the priority dispute?

LeibnizBerkeleyKeillNewton himself

4. Newton’s ẏ/ẋ = 3x² says the same as which statement in Leibniz’s notation?

dy/dx = 3x²∫ y dx = 3x²d²y/dx² = 3x²dx/dy = 3x²

5. Berkeley’s “ghosts of departed quantities” (1734) criticised…

Leibniz’s choice of the letter dtreating increments as non-zero, then as zeroNewton’s theory of coloursthe 1712 committee

6. What did the Analytical Society, founded at Cambridge in 1812, campaign for?

Newton’s dot notationBanning infinitesimalsContinental methods and Leibniz’s d notationA new Royal Society inquiry

7. Archimedes showed a parabolic segment’s area is what multiple of the triangle with the same base and vertex?

4/33/22/3π/2

8. Whose 1821 Cours d’analyse set out the basic theorems of calculus as rigorously as possible?

EulerWeierstrassMaclaurinCauchy

Scoring needs JavaScript turned on. Answer key: 1. A fluxion; 2. 1684; 3. Newton himself; 4. dy/dx = 3x²; 5. treating increments as non-zero, then as zero; 6. Continental methods and Leibniz’s d notation; 7. 4/3; 8. Cauchy.

NotesSources and credits

Every date and claim on this page was checked against the sources below. Where sources give only a decade or a range, so does this page.

J. J. O’Connor and E. F. Robertson, MacTutor History of Mathematics, University of St Andrews: A history of the calculus, on Archimedes’ parabola and series, Newton’s October 1666 tract and method of moments, Barrow’s differential triangle, Leibniz’s 1684 and 1686 papers, the Bernoullis, Berkeley and Maclaurin.
MacTutor biographies: Archimedes; Pierre de Fermat (26 April 1636 letter, method of maxima, minima and tangents); Isaac Newton (plague years, Method of Fluxions 1671/1736, Principia 1687, Keill, the committee and Commercium epistolicum); Isaac Barrow; Gottfried Wilhelm Leibniz (21 November 1675 manuscript, Nova Methodus, Jacob Bernoulli’s “enigma”, Charta volans); John Keill; George Berkeley (The Analyst and the “ghosts” quotation); Charles Babbage (Analytical Society, Lacroix translation, the “dots” quotations); Augustin-Louis Cauchy; Karl Weierstrass (1859–60 lectures).
David Brewster, Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton, vol. 2 (1855), chapter XV, at The Newton Project, for Keill’s letter in the Philosophical Transactions for 1708, the committee’s dates (6 March and 24 April 1712), its finding, and the report being in Halley’s handwriting.
Stanford Encyclopedia of Philosophy, Isaac Newton, on the October 1666 tract and Newton’s early mathematical work.
J. Stachel, introduction to Einstein’s Miraculous Year (Princeton University Press, 2005), sample chapter, quoting Newton’s memorandum on the plague years.

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Newton and Leibniz: The Story of Calculus

Section IA timeline of calculus

c. 225 BC: Archimedes squares the parabola

1630s: Fermat: maxima, minima and tangents

1665–66: Newton’s plague years

1670: Barrow’s Lectiones Geometricae

1675: Leibniz writes ∫ for the first time

1684: Nova Methodus is published

1687: Newton’s Principia

1712: The Royal Society rules for Newton

1734: Berkeley’s The Analyst

1821: Cauchy’s Cours d’analyse

1859–60: Weierstrass and the foundations