Before Calculus: The Problems No One Could Solve
By the mid-17th century, European mathematicians had hit a wall. They could calculate areas of polygons and circles, but the area beneath a curved line — such as the path of a cannonball — was beyond them. They could compute average speed over a journey, but not the exact speed at one instant. And they could not explain, mathematically, why planets moved in ellipses rather than straight lines.
These were not academic puzzles. Navigators needed precise planetary positions for celestial navigation. Engineers needed to understand forces and motion. Physics needed a language for describing the world dynamically — not just statically. The mathematical tools needed to answer these questions would become calculus.
The central challenge: how do you find the slope of a curve at a single point (not a line between two points)? And how do you find the exact area under a curved line? Both require reasoning about infinitely small quantities — and no existing mathematics could handle that.
Isaac Newton: The Method of Fluxions
Isaac Newton (1643–1727) developed his calculus between 1665 and 1666 during what historians call his "annus mirabilis" — his miraculous year. Cambridge University had closed due to the Great Plague, and Newton retreated to his family farm in Lincolnshire. In isolation, he invented calculus, his law of universal gravitation, and the foundations of optics — all in roughly eighteen months.
Newton called his calculus the "method of fluxions." He thought of quantities as flowing over time — a "fluent" was a changing quantity, and a "fluxion" was its rate of flow. In modern terms, a fluxion is a derivative. Newton used the notation ẋ (x with a dot above it) to denote the fluxion of x — a notation physicists still use today for time derivatives.
Newton wrote the derivative of x with respect to time as ẋ (x-dot). The second derivative was ẍ (x-double-dot). This dot notation is still standard in physics — you will see it in mechanics and differential equations courses.
Newton did not publish his method of fluxions for decades. He circulated manuscripts among a small circle of colleagues, but his major works on calculus were not published until 1704 (as an appendix to his Opticks) and 1736 (posthumously). This delay would fuel the great priority dispute.
Gottfried Wilhelm Leibniz: A Cleaner Notation
Gottfried Wilhelm Leibniz (1646–1716) developed his calculus independently, beginning around 1675. Where Newton was driven by physics, Leibniz approached mathematics more abstractly — he was searching for a universal symbolic language that could represent all of human thought. His calculus was part of that grander project.
Leibniz was a far more careful notational thinker than Newton. He agonised over how to write calculus symbolically, and by 1675 he had arrived at the notation we still use today:
∫ f(x) dx — the integral of f(x) (∫ is a stylised 'S' for summa)
Leibniz published his calculus in 1684 and 1686 in the journal Acta Eruditorum — years before Newton's calculus appeared in print. This would become one of the key facts in the dispute: Leibniz published first, even if Newton had developed his ideas first.
Timeline of Calculus
The Bitter Priority Dispute
The dispute over who invented calculus is one of the most famous controversies in the history of science. On one side: Isaac Newton and his British supporters, who argued that Leibniz had seen Newton's unpublished manuscripts during a visit to London in 1676 and borrowed key ideas. On the other: Leibniz and his Continental European supporters, who maintained that his calculus was developed entirely independently.
In 1713, the Royal Society of London published a report called the Commercium Epistolicum that effectively declared Newton the sole inventor and accused Leibniz of plagiarism. The investigation was deeply biased — Newton himself had anonymously written large portions of the report. Leibniz died in 1716, largely discredited in England, though celebrated on the Continent.
Newton vs Leibniz: A Comparison
The Legacy: Whose Notation Won?
Leibniz won the notation war decisively. His dy/dx for derivatives and ∫ for integrals are used universally in mathematics textbooks, research papers, and computer algebra systems worldwide. Newton's dot notation survives specifically in physics, where ẋ for the time derivative of position (velocity) and ẍ for acceleration remain standard.
Why did Leibniz's notation win? Because it is more algebraically intuitive. You can manipulate dy/dx almost like a fraction — cancelling "d"s in the chain rule: dy/dx = (dy/du)(du/dx). This suggestive algebraic behaviour made Leibniz's notation far easier to teach and use, and Continental European mathematicians who used it rapidly outpaced their British counterparts who stuck with Newton's fluxions.
Britain's loyalty to Newton's notation caused its mathematicians to fall behind Continental Europe for over a century. By the time British universities switched to Leibniz's notation in the 1820s, France and Germany had made enormous advances in analysis, differential equations, and mathematical physics.
- Boyer, C.B. (1959). The History of the Calculus. Dover.
- Bardi, J.S. (2006). The Calculus Wars. Thunder's Mouth Press.
- Guicciardini, N. (2003). Newton's Method and Leibniz's Calculus. Springer.