Leibniz Newton Calculus Controversy: Timeline and Verdict

What Was The Leibniz Newton Calculus Controversy?

The Leibniz Newton calculus controversy was a priority dispute over who invented calculus: Isaac Newton (1643 to 1727, England), who developed his method of fluxions in 1665 and 1666 but published little of it, or Gottfried Wilhelm Leibniz (1646 to 1716, Germany), who developed his differential calculus from 1675 and published it first, in 1684. Supporters of Newton accused Leibniz of plagiarism, the quarrel peaked around 1712, and today's consensus is that the two men discovered calculus independently.

Both men were solving the same two problems, rates and areas, and both saw that one undoes the other. In modern terms, they each found the derivative and the integral and the link between them:

$$\frac{d}{dx}\int_{a}^{x} f(t),dt = f(x).$$

Here $f$ is a continuous function, $a$ is a fixed starting point, and the integral measures the area under $f$ from $a$ to $x$. This is the fundamental theorem of calculus. Newton reached it starting from rates of change; Leibniz reached it starting from sums. The full history of the subject, from Archimedes onward, is on the history of calculus page; this article covers the dispute itself.

Who Invented Calculus First, Newton Or Leibniz?

Newton invented his version of calculus first, and Leibniz published first. Both statements are true, and the whole controversy comes from the gap between them.

The dates, checked against the MacTutor biographies of both men:

Newton's lead was real. He had a working calculus of tangents and areas by 1666, nearly a decade before Leibniz's first notes, but he kept it in manuscript and letters. Leibniz, working from 1675, published in 1684, three years before Newton's Principia and long before Newton printed his fluxion notation (partly in 1693, more fully in 1704).

How Did Newton And Leibniz Correspond?

Newton and Leibniz corresponded by letter, routed through Henry Oldenburg, secretary of the Royal Society, and the exchange of 1676 and 1677 was polite. Leibniz visited London in early 1673 and passed through again in October 1676, while Newton held the Lucasian chair of mathematics at his university from 1669.

The key letters:

• 13 June 1676 (Newton). A letter describing his results on infinite series, including the binomial series.

• 24 October 1676 (Newton). A second, longer letter; it did not reach Leibniz until June 1677, by which time he was in Hanover.

• 1677 (Leibniz). A reply in which, according to Wikipedia's account of the dispute, Leibniz used his differential notation.

In the October letter Newton remarked that Leibniz had developed several methods, one of them new to Newton. At that point each man could see the other was far along, and the exchange was polite.

How Did The Dispute Escalate From 1699 To 1716?

The dispute escalated through a chain of accusations, each one pulling Newton and Leibniz further in, until the Royal Society issued a one-sided ruling. It began with Newton's supporters, not with Newton himself.

The escalation, step by step:

1. 1699. The Swiss mathematician Nicolas Fatio de Duillier publicly accused Leibniz of plagiarising Newton.

2. 1704. An anonymous review of Newton's tract on quadrature implied that Newton had borrowed the fluxional calculus from Leibniz. Newton's circle attributed the review to Leibniz.

3. 1708. John Keill accused Leibniz of plagiarism in the Royal Society's Philosophical Transactions. Leibniz read the paper in 1711 and demanded a retraction.

4. 1711. Leibniz wrote to the Royal Society asking it to correct the wrong done to him by Keill's claims.

5. 1712. The Royal Society, with Newton as its president since 1703, set up a committee to pronounce on priority. It never asked Leibniz for his version.

6. 1713. The committee's report, written by Newton himself and published as the Commercium Epistolicum (dated 1712), circulated near the beginning of 1713. It found for Newton. Leibniz did not see it until autumn 1714.

Newton then wrote an anonymous review of the report in the Philosophical Transactions. Leibniz never accepted the verdict, never finished his own history of the calculus, and died in Hanover on 14 November 1716 with the dispute unresolved.

What Does The Evidence Show About Plagiarism?

The evidence shows that Leibniz saw some of Newton's work but developed his calculus by his own route, which is why the modern consensus is independent discovery. The plagiarism charge in the historical dispute was always aimed at Leibniz, never at Newton.

The evidence, weighed on both sides:

• Against Leibniz. In 1849 C. I. Gerhardt found, among Leibniz's papers, extracts from Newton's De Analysi copied in Leibniz's handwriting, with some content re-expressed in Leibniz's notation. Leibniz also admitted, shortly before his death, that Collins had shown him some of Newton's papers in 1676.

• For Leibniz. His private notebooks show his own route to calculus: he started from integration as a generalised summation of series, while Newton started from rates of change. His notation, which Newton never used, made the method a general algorithm.

• The unknown. No document shows when Leibniz made the De Analysi extracts. If he made them in 1675 or 1676, they could have helped him; if he made them from the printed edition of 1704, they could not.

The historian A. Rupert Hall summed it up in 1980, as quoted in Wikipedia's account: Newton first devised the new calculus and understood its potential, and Leibniz created the differential and integral calculus independently. That is where almost every modern history now lands.

How Did Newton's Fluxions Differ From Leibniz's Differentials?

Newton's fluxions and Leibniz's differentials give the same results through different pictures of change. Newton imagined quantities (fluents) flowing in time and measured their speeds (fluxions); Leibniz imagined quantities changing by infinitely small differences.

The table hides the key practical difference. Leibniz's symbols carry the variables with them, so rules such as the chain rule look like fraction algebra: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Newton's dots record a rate with respect to time but do not show which variable you are differentiating with respect to, which makes long calculations harder to check. In 1696 Guillaume de l'Hôpital, writing the first calculus textbook from Leibniz's viewpoint, acknowledged Newton's work but preferred Leibniz's notation for its convenience.

What Were The Consequences Of The Controversy?

The biggest consequence was a split: British mathematicians kept Newton's dot notation and methods, while continental Europe built on Leibniz's, and the two communities drifted apart for about a century. Wikipedia's history of calculus describes the rift as lasting over a century, closed only when Leibnizian calculus was accepted in England in the 1820s.

Three effects stand out:

• A notation gap. Continental mathematicians, including Leibniz's ally Johann Bernoulli, developed calculus in Leibniz's $d$ notation, while British work in fluxions became harder to connect to it.

• A deliberate repair. In 1812 a group of English undergraduates, including Charles Babbage, John Herschel and George Peacock, founded the Analytical Society. Babbage later joked that it existed "for the propagation of d's" against "the heresy of dots". In 1816 they published an English translation of Lacroix's calculus textbook in Leibniz notation.

• Today's mixed notation. Your textbook uses Leibniz's $\frac{dy}{dx}$ and $\int$ for most work, Newton's $\dot{x}$ for time derivatives in physics, and Lagrange's prime $f'(x)$ for functions.

The dispute also left a lesson about evidence. Newton had won the official ruling, yet once both men's manuscripts were published and dated, the ruling lost its authority because the committee had heard only one side.

Why Did Priority Matter So Much In The Seventeenth Century?

"Priority was the only proof of credit" is the short answer. Scientific journals were only beginning, so there was no accepted way to show who discovered something first.

Scientists protected discoveries in ways that look strange today:

• Letters to a secretary. A letter to Henry Oldenburg of the Royal Society, or to Marin Mersenne in Paris, served almost as a publication, because it dated the idea.

• Anagrams and sealed envelopes. A discoverer could hide a result in scrambled letters or a sealed note, then reveal it later to prove an early date.

• Credit for the best use. A later, more useful publication could outweigh an earlier private discovery, which is exactly the argument Leibniz's side made.

Examples Of The Leibniz Newton Calculus Controversy

Each example works a problem the way Newton or Leibniz would have, then states it in modern notation. The point is that both methods always give the same answer.

Example 1

Use Newton's method of moments to find the rate of change of $y = x^3 + 2x$.

Newton replaced $x$ with $x + o$, where $o$ is a vanishingly small increment:

$$(x + o)^3 + 2(x + o) - \left(x^3 + 2x\right) = 3x^2 o + 3x o^2 + o^3 + 2o$$

Divide by $o$:

$$3x^2 + 2 + 3x o + o^2$$

Newton then struck out the terms still containing $o$, because they are "nothing in respect to the rest":

$$3x^2 + 2$$

At $x = 2$ the rate is $3 \cdot 4 + 2 = 14$.

Final answer: $\frac{dy}{dx} = 3x^2 + 2$, which equals $14$ at $x = 2$.

Example 2

Use Leibniz's rule $d\left(x^n\right) = n x^{n-1},dx$, which he had by autumn 1676, to find $dy$ for $y = x^5$.

$$dy = d\left(x^5\right)$$

$$= 5x^4,dx$$

Dividing by $dx$ gives the derivative $\frac{dy}{dx} = 5x^4$, the power rule in its modern form.

Final answer: $dy = 5x^4,dx$.

Example 3

Differentiate $y = \left(x^2 + 1\right)^3$ and evaluate the result at $x = 1$.

Working only from "bring down the power" in Newton's dot style, a first attempt writes

$$\dot{y} = 3\left(x^2 + 1\right)^2$$

and at $x = 1$ gets $3 \cdot 4 = 12$.

Check that against a direct estimate: from $x = 1$ to $x = 1.001$, $y$ grows from $8$ to about $8.024$, a rate of about $24$, double the attempt. The rule was applied to the outer cube but not to the inside, which changes too. Leibniz's notation makes the missing piece visible. Let $u = x^2 + 1$:

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

$$= 3u^2 \cdot 2x$$

$$= 6x\left(x^2 + 1\right)^2$$

At $x = 1$:

$$6 \cdot 1 \cdot 4 = 24$$

In correct fluxion form the same answer reads $\dot{y} = 6x\left(x^2 + 1\right)^2 \dot{x}$, with the $\dot{x}$ carrying the inner rate.

Final answer: $\frac{dy}{dx} = 6x\left(x^2 + 1\right)^2$, which equals $24$ at $x = 1$.

Example 4

Use Newton's area rule from De Analysi, that the area under $y = x^n$ from $0$ to $x$ is $\frac{x^{n+1}}{n + 1}$, to find the area under $y = \sqrt{x}$ from $0$ to $4$.

Newton allowed fractional powers, so $\sqrt{x} = x^{1/2}$ and $n = \frac{1}{2}$:

$$\text{Area} = \frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2}$$

$$\text{At } x = 4: \quad \frac{2}{3} \cdot 8 = \frac{16}{3}$$

In Leibniz's notation the same calculation is the definite integral

$$\int_{0}^{4} \sqrt{x},dx = \frac{16}{3}$$

Newton then checked such results by differentiating the area and recovering the curve, the fundamental theorem in action: $\frac{d}{dx}\left(\frac{2}{3}x^{3/2}\right) = \sqrt{x}$.

Final answer: the area is $\frac{16}{3}$.

Example 5

Let $x = t^2$ and $y = t^3$. Find the rate of change of $xy$ with respect to $t$ in both notations.

Leibniz's product rule (the product rule):

$$d(xy) = x,dy + y,dx$$

$$\frac{d(xy)}{dt} = t^2 \cdot 3t^2 + t^3 \cdot 2t$$

$$= 5t^4$$

Newton's fluxions:

$$\dot{(xy)} = x\dot{y} + y\dot{x}$$

$$= t^2 \cdot 3t^2 + t^3 \cdot 2t = 5t^4$$

Direct check: $xy = t^5$, whose rate is $5t^4$.

Final answer: $5t^4$ in both notations.

Example 6

The radius of a circle grows at $2$ cm per second. How fast is the area growing when the radius is $5$ cm?

Newton would treat $A$ and $r$ as fluents in time: $A = \pi r^2$ gives $\dot{A} = 2\pi r,\dot{r}$. Leibniz would write the same thing as a related rates equation:

$$\frac{dA}{dt} = 2\pi r,\frac{dr}{dt}$$

$$= 2\pi \cdot 5 \cdot 2$$

$$= 20\pi$$

To 4 decimal places that is $62.8319$ square centimetres per second.

Final answer: $20\pi \approx 62.8319,\text{cm}^2/\text{s}$.

Common Errors About The Leibniz Newton Calculus Controversy

The errors below come from mixing up discovery with publication, and from treating two notations as two different subjects.

Mistake 1: Saying Leibniz Stole Calculus

Where it slips in: Quick summaries and quiz answers that repeat the 1713 verdict as fact.

Don't do this: Write "Leibniz copied Newton" because the Royal Society said so.

The correct way: Say both discovered calculus independently, and note that the Royal Society's report was written by Newton and never heard Leibniz's side.

The memorizer who learned the verdict as a date and a winner gets caught here. A student who knows the committee was Newton's own sees at once why historians set its ruling aside.

Mistake 2: Equating First Published With First Discovered

Where it slips in: Questions that ask "who invented calculus first" and expect one name.

Don't do this: Answer "Leibniz, 1684" or "Newton, 1687" as if publication dates settled the question.

The correct way: Separate the two: Newton discovered first (1665 to 1666), Leibniz published first (1684), and Newton's own calculus appeared in print only partly in 1693 and more fully in 1704.

Mistake 3: Treating Fluxions And Differentials As Different Mathematics

Where it slips in: Physics problems that switch between $\dot{x}$ and $\frac{dx}{dt}$.

Don't do this: Expect a different answer depending on the notation, or drop the inner rate $\dot{x}$ when using dots.

The correct way: Translate directly: $\dot{x}$ means $\frac{dx}{dt}$, and every fluxion result has a Leibniz twin, as Examples 3 and 5 show.

The rusher drops the $\dot{x}$ in a chain-rule step because the dot is small and easy to skip. Students who write the variable of differentiation every time, as Leibniz's $\frac{dy}{dx}$ forces you to, stop losing that inner factor.

Practice Problems On The Leibniz Newton Calculus Controversy

1. Use Newton's method of moments (replace $x$ by $x + o$, divide by $o$, strike out the terms with $o$) to find the rate of change of $y = x^4 - 3x$.

2. Use Leibniz notation with $u = x^2$ to differentiate $y = \sin\left(x^2\right)$.

3. Evaluate $\int_{0}^{2} x^3,dx$.

4. Use Newton's area rule for fractional powers to find the area under $y = x^{1/3}$ from $0$ to $8$.

5. A sphere's radius grows at $\frac{1}{2}$ cm per second. How fast is its volume $V = \frac{4}{3}\pi r^3$ growing when $r = 3$ cm?

6. Rewrite Newton's fluxion equation $\dot{y} = 3x^2,\dot{x}$ in Leibniz notation.

Answer to Question 1: $4x^3 - 3$.

Answer to Question 2: $\frac{dy}{dx} = \cos(u) \cdot 2x = 2x\cos\left(x^2\right)$.

Answer to Question 3: $\frac{2^4}{4} = 4$.

Answer to Question 4: $\frac{3}{4} \cdot 8^{4/3} = \frac{3}{4} \cdot 16 = 12$.

Answer to Question 5: $\frac{dV}{dt} = 4\pi r^2 \cdot \frac{1}{2} = 18\pi$ cubic centimetres per second.

Answer to Question 6: $\frac{dy}{dt} = 3x^2,\frac{dx}{dt}$, or $\frac{dy}{dx} = 3x^2$.

The Short Version

• The Leibniz Newton calculus controversy was a priority dispute, and the modern verdict is independent discovery.

• Newton discovered his method of fluxions in 1665 and 1666; Leibniz developed his calculus from 1675 and published first, in 1684.

• The 1712 Royal Society committee ruled for Newton, but Newton wrote its report and Leibniz was never heard.

• British loyalty to Newton's dot notation separated British and continental mathematics until the Analytical Society's work from 1812.

• Fluxions and differentials always give the same answers; Leibniz's notation simply makes the bookkeeping easier.

Your Next Move

Work through practice problems 1 to 3 in both Newton's and Leibniz's notation, then try 4 to 6; if the chain rule in problem 2 causes trouble, return to Example 3 and the notation table. For a live session that works through both notations side by side, book a free demo with a Bhanzu trainer:

Read More

• What is calculus

• Table of derivatives

• Integration by substitution

• Power series

• Derivative as rate of change