Leibniz writes the integral sign, giving calculus its lasting language

Story: 1675 C.E.  |  Last updated: October 29, 1675

On a single page of his private notebook in the autumn of 1675 C.E., a 29-year-old German mathematician drew a long, curved symbol beside a string of variables and changed how humans would think about continuous change forever. The symbol was the integral sign — still used in every calculus textbook on Earth today.

What the evidence shows

• Leibniz integral calculus: On October 29, 1675 C.E., Gottfried Wilhelm Leibniz first wrote the elongated “∫” symbol — derived from the Latin summa, meaning “sum” — in his working notebooks, marking the first recorded use of integral notation in the form mathematicians still use.
• Differential notation: In the same period, Leibniz developed the “d” notation for differentials (as in dx, dy), creating a complete symbolic language for calculus that was cleaner and more generalizable than the geometric methods Isaac Newton had used in parallel.
• Independent discovery: Newton and Leibniz developed their versions of calculus independently and nearly simultaneously — Newton beginning around 1666 C.E., Leibniz reaching his key notation by 1675 C.E. and publishing first, in 1684 C.E.

A mind that refused to stay in one lane

Leibniz was not a specialist. Born in Leipzig in 1646 C.E., he earned degrees in philosophy and law before turning seriously to mathematics. He taught himself much of what he knew about advanced math during a diplomatic visit to Paris in the early 1670s C.E., where he encountered the work of Dutch mathematician Christiaan Huygens.

That autodidactic energy showed in everything he touched. Within a few years of arriving in Paris, Leibniz had developed a working mechanical calculator capable of multiplication and division, sketched ideas that would later become binary arithmetic, and begun drafting the symbolic framework for calculus. He was doing all of this while also writing philosophy, working in diplomacy, and corresponding across Europe.

What made the 1675 C.E. notebook entries so significant was not just the mathematics — it was the notation. Mathematics lives and dies by its symbols. A good notation doesn’t just record an idea; it makes previously impossible ideas easy to think. Leibniz understood this intuitively. His integral sign and differential notation were so well-designed that the mathematical community eventually adopted them wholesale over Newton’s more geometric, dot-based system.

Why notation matters more than it seems

It can be tempting to treat mathematical symbols as mere shorthand — convenient labels for things that exist independently of how we write them. But notation shapes thought. The way Leibniz wrote calculus made it easier to manipulate, extend, and apply. His notation invited generalization.

Within decades of his 1684 C.E. publication, mathematicians across Europe — including the Bernoulli brothers in Switzerland — were using Leibniz’s system to solve problems in physics, engineering, and astronomy that had been intractable before. The notation traveled. The ideas multiplied.

Today, when an engineer calculates the stress on a bridge, when a physicist models how heat moves through a material, when an economist builds a model of diminishing returns, they reach for the same ∫ that Leibniz scratched into a notebook in 1675 C.E. The symbol is now so universal that most people who use it have no idea where it came from.

Calculus as a convergence, not a conquest

The famous priority dispute between Leibniz and Newton — which became bitter, nationalistic, and professionally damaging to Leibniz in his final years — obscures something more interesting: two people, working in different countries with different methods, converged on the same mathematical truth at roughly the same moment.

That convergence was not a coincidence. Both men were working in a tradition that stretched back through Bonaventura Cavalieri’s method of indivisibles in Italy, Pierre de Fermat’s work on tangents in France, and further still to Archimedes’ exhaustion method in ancient Greece. Indian mathematicians of the Kerala school, notably Madhava of Sangamagrama, had independently developed infinite series expansions for trigonometric functions as early as the 14th century C.E. — work that prefigured calculus in ways European mathematicians were unaware of.

Leibniz and Newton did not create calculus from nothing. They gathered, synthesized, and formalized a century of accumulated European mathematical insight — and, unknowingly, stood on even older shoulders from outside Europe. What Leibniz added, crucially, was a language clear enough to let others continue the work.

Lasting impact

It is almost impossible to overstate how much of the modern world runs on calculus. Classical mechanics, electromagnetism, thermodynamics, and quantum physics are all expressed in the language Leibniz helped create. Engineering, economics, statistics, computer graphics, climate modeling — each depends on differential and integral calculus at its foundation.

Leibniz’s specific contribution — the notation — turned out to be as important as the mathematics itself. His framework made calculus portable. It could be written down, taught, shared, and built upon by people who had never met him and never would. That portability is why the Industrial Revolution could be mathematized, why 19th-century physics became so powerful, and why today’s machine learning algorithms can optimize themselves using gradient descent — a direct descendant of differential calculus.

Leibniz also envisioned, decades before anyone built it, a machine that could reason symbolically. His dream of a calculus ratiocinator — a universal logical calculator — is now recognized as a conceptual ancestor of symbolic computing and, in a broader sense, of computer science itself.

Blindspots and limits

The foundational concepts Leibniz used — infinitely small quantities called “infinitesimals” — were logically shaky by the standards of the time and remained so for nearly two centuries, drawing sharp criticism from philosophers including George Berkeley. It took the work of Augustin-Louis Cauchy and Karl Weierstrass in the 19th century C.E. to place calculus on a rigorous footing through the theory of limits. Leibniz’s notation was brilliant, but the reasoning underneath it was not fully secured until long after his death.

The bitter dispute with Newton also left a real scar: British mathematicians, loyal to Newton’s notation, largely isolated themselves from Continental advances for much of the 18th century C.E. — a reminder that even the most collaborative-seeming intellectual breakthroughs can become entangled in pride, nationalism, and institutional politics.

Read more

For more on this story, see: Wikipedia — Gottfried Wilhelm Leibniz: Calculus

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