How Calculus Was Invented Twice (and Why People Got Mad)

Calculus is one of those inventions that feels inevitable-like once you need it, it simply must appear. And historically, it kind of did. In the late 1600s, two brilliant minds-Isaac Newton in England and Gottfried Wilhelm Leibniz in continental Europe-each built a version of calculus. Independently. Almost simultaneously.

Then everyone got mad.

Not just “academic disagreement” mad. We’re talking national pride, personal accusations, secretive manuscripts, committees, and a feud that shaped the way we still write math today.

Two problems that begged for calculus

Before calculus had a name, it was a pair of stubborn questions:

• Tangent problem: Given a curve, what’s the slope of the tangent at a point? (Think: instantaneous speed from a position curve.)

• Area problem: What’s the area under a curve? (Think: distance traveled from a velocity curve.)

People had partial tools-geometry, algebra, clever tricks-but the “general machine” wasn’t there yet. The Scientific Revolution was heating up: planetary motion, optics, mechanics. Everyone needed a way to handle continuous change.

Newton and Leibniz both built that machine… in their own styles.

Newton’s “fluxions”: calculus as motion

Newton thought like a physicist (because he basically invented half of physics while he was at it). For him, quantities weren’t static numbers-they were things that flowed.

• A changing quantity was a fluent.

• Its rate of change was a fluxion.

In modern language: if $x(t)$ changes over time, Newton wanted a notation for $\frac{dx}{dt}$. He used dots:

$\dot{x} \quad \text{means} \quad \frac{dx}{dt}$

This is why, in physics today, time derivatives often still wear Newton’s dots.

Newton developed his methods in the mid-1660s (yes, that plague-era “working from home” period), but he was famously reluctant to publish. He shared ideas in letters and private notes, but he didn’t put a clean, public “here is calculus” stamp on it right away.

Leibniz’s differential calculus: calculus as symbols

Leibniz approached the same mountain from a different trail. He loved notation, structure, and symbolic clarity. He introduced:

• Differentials $dx$and$dy$: tiny changes

• The derivative as a ratio:

$\frac{dy}{dx}$

And for accumulation (the reverse process), he introduced the elongated $S$:

$\int f(x)\,dx$

Leibniz published first, in 1684 (derivatives) and 1686 (more development). His notation was so usable that it spread rapidly across Europe-especially through the Bernoulli brothers and later Euler, who basically turned calculus into a superpower.

So even if Newton had the earlier ideas, Leibniz had the earlier publication and the stickier notation.

And then… drama.

So who “really” invented calculus?

Here’s the cleanest honest summary:

• Newton developed his core ideas earlier (mid-1660s), mostly privately.

• Leibniz published earlier (1680s), clearly and systematically.

• Their approaches were similar in power but different in style: Newton $=$ time and motion; Leibniz $=$ symbols and general methods.

The conflict wasn’t “did Newton do it?” or “did Leibniz do it?” It was: did Leibniz copy Newton?

Newton’s supporters believed yes. Leibniz insisted no.

And because intellectual credit mattered (and national rivalries didn’t help), things escalated.

Why people got mad: priority, pride, and a committee with a thumb on the scale

In the early 1700s, accusations of plagiarism began circulating. The Royal Society got involved. A committee investigated.

Problem: Newton was effectively running the show behind the scenes, and the committee report (the famous Commercium Epistolicum) leaned hard toward Newton.

The result was a long-lasting split:

• British mathematicians often stuck with Newton’s fluxional notation and style.

• Continental mathematicians ran with Leibniz’s notation and developed calculus at high speed.

Historically, Britain’s isolation from the Leibniz/Euler style arguably slowed British math for a while. (Not because British mathematicians weren’t talented, but because notation and shared methods matter-a lot.)

And that’s the hidden lesson: notation isn’t decoration. It’s infrastructure.

A tiny worked example: the same derivative, two “languages”

Let’s do one simple derivative-something you can feel in your hands.

Take:

$y = x^2$

Leibniz-style (what you learned in school)

Start from the idea of a small change. If $x$ changes by a tiny amount $dx$, then:

$y = x^2 \quad \Rightarrow \quad y + dy = (x + dx)^2$

Expand:

$y + dy = x^2 + 2x\,dx + (dx)^2$

Subtract $y=x^2$:

$dy = 2x\,dx + (dx)^2$

Divide by $dx$:

$\frac{dy}{dx} = 2x + dx$

Now comes the calculus move: when dx is “tiny,” the extra $dx$ term becomes negligible compared with the constant $2x$. So:

$\frac{dy}{dx} = 2x$

That’s the derivative.

Newton-style (motion-first intuition)

Newton imagines $x$ depends on time: $x=x(t)$. Then $y=x^2$ also depends on time. Differentiate with respect to time:

$y = x^2 \quad \Rightarrow \quad \dot{y} = 2x\,\dot{x}$

If you want the rate of change of $y$ per change in $x$, you divide:

$\frac{\dot{y}}{\dot{x}} = 2x$

Same result. Different viewpoint.

Leibniz hands you a flexible symbolic tool for “change with respect to x.”

Newton starts from time and physics, then backs into slope.

Both are calculus. Both are brilliant.

The real winner: the Fundamental Theorem of Calculus

The biggest “mic drop” of early calculus is that the two big problems-tangents and areas-are secretly reverse operations.

In modern notation:

• If $F'(x) = f(x)$, then

$\int_a^b f(x)\,dx = F(b) - F(a)$

This is why calculus feels like a machine:

• derivatives turn accumulation into instantaneous rate

• integrals turn instantaneous rate into accumulation

• and the theorem stitches them together.

Neither Newton nor Leibniz just did a party trick. They built a bridge between two worlds.

Why Leibniz’s notation won (even in England… eventually)

Leibniz notation has a few unfair advantages:

• $\frac{dy}{dx}$ visually screams “rate of change of $y$ with respect to $x$”

• $\int f(x)\,dx$ looks like a sum of infinitesimals (which matches intuition)

• it generalizes cleanly: partial derivatives, multivariable calculus, differential forms

Newton’s dot notation is perfect for time derivatives in mechanics, but it doesn’t scale as cleanly when you start differentiating with respect to other variables.

So history did what history does: it kept what traveled best.

But Newton’s approach still lives in physics, engineering, and differential equations-anywhere time is the natural driver.

The takeaway: calculus wasn’t just discovered-it was negotiated

The Newton–Leibniz feud is entertaining, but the deeper point is this:

• Big ideas often arrive when the world is ready for them.

• Credit depends on communication, not only insight.

• Notation can decide the future of a field.

Calculus wasn’t merely invented twice. It was adopted once-through the language that made it easiest to teach, share, and extend.

And yes… people got mad because the stakes were enormous: prestige, priority, and the right to claim ownership of the mathematical engine powering modern science.

Key takeaways

• Newton and Leibniz both created calculus, with different motivations and notation.

• Newton developed earlier but published later; Leibniz published earlier and spread faster.

• The feud was largely about priority, plagiarism accusations, and national rivalry.

• Leibniz’s notation won broadly because it’s flexible and scalable.

• A single derivative like $y=x^2$ shows how both “languages” reach the same truth.