Newton and Leibniz — The Birth of Calculus and a 30-Year Priority Dispute

The Summit of the 17th-Century Revolution — Calculus

Building on the coordinate geometry and maxima-minima methods covered in the previous article, the second half of the 17th century sees the emergence of calculus — an integrated mathematical instrument. Its two independent inventors are Newton (1643–1727) and Leibniz (1646–1716).

• Discovery — Newton first (1665–66 vs. 1675)
• Publication — Leibniz first (1684 vs. 1704)
• Notation — Leibniz decisive winner (the still-used ∫ and dy/dx)
• Political victor — Newton, who controlled the Royal Society
• Historical verdict — independent discovery, now consensus

Their discoveries, dispute, and legacies built the basic apparatus of modern mathematics.

Isaac Newton — Genius of the “Annus Mirabilis”

Isaac Newton (1643–1727) was an English mathematician, physicist, astronomer, and natural philosopher. His physical side (gravitation, laws of motion, optics) sits alongside his mathematics; this article focuses on Newton the mathematician.

Life

• 4 January 1643 — Born at Woolsthorpe
• 1661 — Enters Trinity College, Cambridge; attends Barrow’s lectures
• 1664–65 — Self-teaches from Descartes’s Geometry and Wallis’s Arithmetica Infinitorum
• 1665–66 — Cambridge closed by plague, returns home. The annus mirabilis — calculus, gravitation, optics all conceived
• 1669 — De Analysi per Aequationes Numero Terminorum Infinitas completed. Appointed Lucasian Professor at 26
• 1687 — Publishes the Principia
• 1704 — Publishes De Quadratura Curvarum and Enumeratio Linearum Tertii Ordinis as appendices to the Opticks (his first formal publications on calculus)
• From 1711 — Organizes a Royal Society “commission” that in fact he himself directs, and denounces Leibniz
• 20 March 1727 — Dies; buried in Westminster Abbey

The Annus Mirabilis, 1665–66

Over roughly 18 months, during which plague had closed Cambridge, the 22–23-year-old Newton at Woolsthorpe conceived:

• Calculus — the foundations of his “method of fluxions”
• Universal gravitation — the “falling apple” (the truth of the story is disputed)
• Optics — decomposition of white light by prism; spectrum theory

“A single 23-year-old, in the countryside for 18 months, built the whole of modern science” — a symbolic moment in the history of science. The mathematics and physics would be refined for decades, but the seeds were sown here.

The Method of Fluxions — Newton’s Calculus

Newton’s calculus is called the method of fluxions:

• Fluent — a quantity that varies with time (the modern function y(t))
• Fluxion — its rate of change (modern dy/dt), written ẏ (y-dot)
• Inverse method — reconstructing the fluent from the fluxion (indefinite integration)

The conception is essentially kinematic: quantities “flow” with time. This meshes directly with the Principia’s physics — Newton invented calculus as the mathematics of gravitation.

Newton’s Mathematical Achievements

The Generalized Binomial Theorem (c. 1665)

Reading Wallis, Newton generalized (1 + x)^n to arbitrary real n (negative, fractional):

(1 + x)^n = 1 + nx + n(n-1)/2! x² + n(n-1)(n-2)/3! x³ + …

For positive integer n, finitely many terms; otherwise an infinite series. This became the starting point of the theory of infinite series.

Infinite Series

Newton systematized power-series expansions for 1/(1 − x) = 1 + x + x² + …, log(1 + x), sin x, cos x, arcsin x, and much more. A foundational toolkit of modern analysis.

Newton’s Method

An iterative numerical solver for f(x) = 0:

x_{n+1} = x_n − f(x_n) / f’(x_n)

Introduced in Newton’s 1669 paper. Today’s Newton–Raphson method is a staple of numerical computation.

Classification of Cubic Curves

In Enumeratio Linearum Tertii Ordinis (1704), Newton classified cubic algebraic curves into 72 types (later one more, 73). A pioneering work in algebraic geometry.

Gottfried Leibniz — The “Universal Genius”

Gottfried Wilhelm Leibniz (1646–1716) was a German philosopher, mathematician, scientist, and diplomat. Alongside Newton, he was the independent inventor of calculus, and introduced the notation ∫ (integral) and d/dx that we still use.

He also anticipated binary arithmetic, mechanical calculation, formal logic, and combinatorics — many of the core ideas of computer science — earning the title of universal genius. His philosophy (monadology, pre-established harmony, “best of all possible worlds”) is in itself an enormous project.

Life

• 1 July 1646 — Born in Leipzig; his father Friedrich was a professor of moral philosophy at Leipzig University
• 1653 — Father dies; the 7-year-old Leibniz teaches himself Latin from his father’s library
• 1661 — Enters Leipzig University at 15; law and philosophy
• 1666 — Publishes De Arte Combinatoria at 20
• 1667 — Doctorate in law from Altdorf
• 1672–76 — Lives in Paris; learns mathematics from Huygens; discovers calculus (c. 1675)
• From 1676 — Serves at the Hanoverian court as diplomat and librarian
• 1684 — Publishes calculus paper in Acta Eruditorum
• 1700 — Founds the Berlin Academy; first president
• 14 November 1716 — Dies in Hanover, aged 70. No one from the court attended his funeral

Breadth

Leibniz was active in philosophy (monadology), law (natural law), diplomacy, theology, history (the Hanoverian genealogy), linguistics (universal language, Chinese characters), mining engineering (Harz drainage), calculating-machine design. Very few individuals in history have contributed at the top level in so many fields.

Leibniz’s Notation — The Winner of Modern Mathematics

Leibniz’s decisive victory in the notation war:

Symbol	Meaning
dy/dx	Derivative
∫ y dx	Integral (an elongated S — for Sum)
d²y/dx²	Second derivative

These are far more operational and algebraic than Newton’s ẏ. Modern mathematics is essentially Leibnizian in its notation — one can “divide dy by dx” intuitively.

The 1684 Paper

The short paper in Acta Eruditorum, “A New Method for Maxima and Minima, and Tangents, which is Impeded neither by Fractional nor by Irrational Quantities,” is the public debut of Leibniz’s differential calculus. Integration followed in 1686. This is the decisive moment when calculus spread across continental Europe.

The Priority Dispute — Royal Society vs. the Continent

The greatest controversy in the history of science at the turn of the 18th century:

Year	Event
1665–66	Newton discovers calculus (unpublished)
1675	Leibniz independently discovers
1684	Leibniz publishes
1693	Leibniz corresponds with Newton
1699	Fatio de Duillier (Newton camp) alleges Leibniz plagiarized
1704	Newton at last publishes on calculus
1708	John Keill (Newton camp) publicly asserts plagiarism
1711	Royal Society commission convened — in fact directed by Newton
1713	Report Commercium Epistolicum rules for Newton
1716	Dispute continues after Leibniz’s death

Modern historians hold that both invented calculus independently. Newton was earlier in discovery (c. 1666 vs. c. 1675); Leibniz was earlier in publication and superior in notation.

A Century of English Stagnation

The dispute split the mathematical community — England vs. the Continent — and English mathematics stagnated through the 18th century (they clung to Newton’s ẏ). On the Continent, Euler, Lagrange, and Laplace developed calculus in Leibniz’s notation.

The break came only in the early 19th century, when Cambridge’s Analytical Society — Babbage, Herschel, Peacock — deliberately switched to Leibniz’s symbols. A rare case in which a notational choice changed a country’s mathematical trajectory by a century.

Leibniz’s Other Contributions — Precursor of Computer Science

Binary (1679–1703)

Leibniz began working on binary arithmetic around 1679 and published Explanation of Binary Arithmetic in 1703. He was fascinated by the correspondence between binary and the hexagrams of the Yi Jing:

• 0, 1, 10, 11, 100, 101, 110, 111 …
• Symbol of divine creation (1 = God, 0 = void) — a theological reading
• Foundation of modern computing

The Step Reckoner

From 1672 Leibniz designed and built a mechanical calculator — the Step Reckoner — capable of all four arithmetic operations (completed c. 1694). This far exceeded Pascal’s machine (addition and subtraction only). Two machines were made; one survives at the Leibniz Library in Hanover.

Precursor of Formal Logic

The Characteristica Universalis — expressing all concepts as symbols and carrying out reasoning by symbolic manipulation:

“When there are disputes among persons, we can simply say: Let us calculate, without further ado, and see who is right.”

Through Boole and Frege in the 19th century, and Russell, Whitehead, Gödel, and Turing in the 20th, this becomes formal logic and computer science.

Determinants (1693)

In a letter to L’Hôpital, Leibniz introduced a construction close to determinants for solving linear systems. About ten years behind Japan’s Seki Takakazu (1683) — but discovered independently.

The Bernoullis — Leibniz’s Disciples

Leibniz spread calculus through the Bernoulli brothers:

• Jakob Bernoulli (1655–1705) — probability, Bernoulli numbers
• Johann Bernoulli (1667–1748) — calculus of variations, the brachistochrone
• L’Hôpital (1661–1704) — his Analyse des Infiniment Petits (1696) is the first textbook of Leibnizian calculus

This lineage runs directly into Euler in the 18th century.

Newton’s Other Face — Alchemy and the Mint

Alongside the mathematician, Newton spent enormous time on alchemy, biblical chronology, and mystical interpretation. Manuscripts recovered after his death showed more than half concerned theology and alchemy. On seeing them, John Maynard Keynes remarked that Newton was “not the first of the age of reason, but the last of the magicians.”

From 1696, Newton served as Warden and then Master of the Royal Mint (1700–27). He worked energetically at coin reform and counterfeit prosecution — and personally pursued cases that led to hangings. “The mathematical giant chasing counterfeiters at the Mint” — a strange late career.

Leibniz’s Lonely End

His last years were politically bleak: when the Elector of Hanover became George I of Britain, Leibniz was not permitted to accompany him to London. No one from the court came to his funeral. He was forgotten for a while, then, from the 19th century onward, rediscovered from three directions — modern philosophy, modern logic, modern mathematics.

The contrast with Newton, buried as a national hero at Westminster Abbey, is stark. But modern computer science descends directly from Leibniz’s binary and his universal characteristic.

From This Article to the Next

We have traced the birth of calculus and the 30-year dispute. The next article turns to the giants who developed calculus explosively — Euler (1707–1783) and Gauss (1777–1855). Euler, who contributed to every branch of mathematics, and Gauss, the “prince of mathematicians” — the two dominant figures of the 18th–19th centuries, and how they made mathematics a genuinely continental discipline.