Chapter 4: Calculus — Newton vs. Leibniz
1665–1716
Newton's Fluxions (1665–1666)
During the annus mirabilis of 1665–66, Isaac Newton, sheltering from plague at Woolsthorpe Manor, developed his “method of fluxions” — what we now call differential and integral calculus. For Newton, a “fluent” was a quantity that flows (changes) in time, and a “fluxion” was its rate of change.
Newton's motivation was entirely physical: he needed calculus to solve the problem of planetary motion. The fundamental theorem of calculus — that differentiation and integration are inverse operations — emerged from his study of areas under curves:
\( \frac{d}{dx}\int_a^x f(t)\,dt = f(x) \)
Newton used his calculus to prove that an inverse-square force law produces elliptical orbits (Kepler's first law), to derive the shell theorem, and to explain tides. Calculus was invented for physics — the purest example of a physical problem creating new mathematics.
Leibniz's Notation (1675–1684)
Gottfried Wilhelm Leibniz independently developed calculus with a fundamentally different approach. Where Newton thought kinematically (quantities flowing in time), Leibniz thought algebraically — manipulating infinitesimal differences \(dx\) and \(dy\).
Leibniz's notation was revolutionary. His symbols \(\frac{dy}{dx}\) for derivatives and \(\int\) for integrals (an elongated “S” for “summa”) are still used universally today. The notation made calculus computable — it suggested the rules:
\( \frac{d(uv)}{dx} = u\frac{dv}{dx} + v\frac{du}{dx} \)
\( \frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx} \) (chain rule)
Why Notation Matters
Continental mathematicians adopted Leibniz's notation and rapidly surpassed British mathematics for a century. The lesson: good mathematical notation is not mere decoration — it shapes what thoughts are possible. The same principle applies to Dirac notation in quantum mechanics and index notation in general relativity.
The Priority Dispute
The Newton–Leibniz priority dispute (1699–1716) was the bitterest scientific controversy in history. Newton accused Leibniz of plagiarism; Leibniz accused Newton of the same. The Royal Society, presided over by Newton, officially ruled in Newton's favor in 1713.
Modern historians agree that both independently invented calculus. Newton was first (1665–66) but published last (1704). Leibniz published first (1684) with superior notation. The dispute isolated British mathematics from Continental developments for over a century.
The Principia: Calculus Meets Gravity
Newton's Philosophiae Naturalis Principia Mathematica (1687) is the most important scientific book ever published. Using his calculus (disguised in geometric form), Newton derived:
- • All three of Kepler's laws from the inverse-square law \(F = \frac{GMm}{r^2}\)
- • The shape of Earth (oblate spheroid)
- • The precession of equinoxes
- • The orbits of comets
- • The theory of tides
For the first time, a single mathematical framework explained both terrestrial and celestial phenomena. The message was unmistakable: the universe is mathematical, and calculus is its language.
The Bernoulli Dynasty
The Bernoulli family of Basel produced eight mathematicians across three generations. Johann Bernoulli (1667–1748) was the most influential teacher of his era — he taught Euler, the greatest mathematician of the 18th century.
Jakob Bernoulli (1654–1705) applied calculus to probability (the law of large numbers). Johann solved the brachistochrone problem — finding the curve of fastest descent under gravity — launching the calculus of variations:
The brachistochrone is a cycloid: \(x = r(\theta - \sin\theta), \; y = r(1-\cos\theta)\). Finding this curve requires minimizing a functional — a “function of functions” — the idea that would become the principle of least action.