Newton

The life of Isaac Newton

Timeline 1643Born on 4 January at Woolsthorpe, England. In English records of the time the date is 25 December 1642. 1661Enters Trinity College, Cambridge, as a poor student. 1665–1667With the university closed by the plague, he returns home. He develops series, fluxions, the theory of colours and his first ideas on gravity. 1669Aged 26, becomes Lucasian Professor of Mathematics, succeeding Isaac Barrow. 1672Presents his reflecting telescope and publishes his theory of light and colours. 1687Publishes the Principia, paid for out of Edmond Halley’s own pocket. 1696Moves to the Royal Mint in London, first as Warden and, from 1699, as Master. 1703Becomes President of the Royal Society. 1704Publishes Opticks. 1727Dies on 31 March in London. He is buried in Westminster Abbey. ◀ Previous Follow the journey Next ▶ 1643–1661 WoolsthorpeHe is born on a farm, the posthumous son of a farmer, and goes to school in nearby Grantham. 1661–1665 CambridgeHe enters Trinity College as a poor student who pays his way by serving other students. 1665–1667 WoolsthorpeWith the university closed by the plague, he goes home. He develops series, fluxions, the theory of colours and his first ideas on gravity. 1667–1696 CambridgeHe holds the Lucasian chair, builds the reflecting telescope and writes the Principia (1687). 1696–1727 LondonHe runs the Royal Mint, at the Tower of London, and presides over the Royal Society. He dies in 1727 and is buried in Westminster Abbey. Enable JavaScript to see the map. The stages are listed below. Newton’s three places, from Woolsthorpe to Cambridge and London. Map: © OpenStreetMap contributors

October 11, 2026 Read

The generalized binomial: beyond whole-number powers

At school we learn to expand $(1+x)^2=1+2x+x^2$ and $(1+x)^3=1+3x+3x^2+x^3$. The coefficients come from the arithmetic triangle, which Pascal studied and which was known long before him. For positive whole-number exponents, the expansion always stops. In the winter of 1664–1665, still a student, Newton asked a question nobody had asked so boldly: what if the exponent were $\frac12$? Or $-1$? What would $(1+x)^{1/2}$, that is $\sqrt{1+x}$, look like as a sum of powers of $x$?

October 11, 2026 Read

Fluxions: how to measure an instantaneous change

A car leaves one city and reaches another, 90 km away, in one hour. Its average speed was 90 km/h. But at every instant the speedometer showed a different number. What exactly does it measure? Speed is distance divided by time, but in an instant no distance is covered and no time passes. Dividing zero by zero gives no answer. That was the question Newton wanted to answer in 1665. His answer was the method of fluxions.

October 11, 2026 Read

Areas and integration: the inverse of fluxions

Measuring the area of curved figures is one of the oldest problems in mathematics. Around 250 BC Archimedes showed that the area of a parabolic segment is $\frac43$ of the area of a triangle inscribed in it. It was a stroke of genius, but made to measure: every new curve needed a new idea. By the 17th century, Fermat, Cavalieri, Wallis and others could already compute the area under $y=x^n$ for many values of $n$. What was missing was a general rule. And what was missing was the realization that Newton and Leibniz each reached in their own way: computing areas and computing rates of change are inverse operations.

October 11, 2026 Read

Newton's method: approximating roots with tangents

Some equations have no formula. There is no quadratic formula for $e^x+x-3=0$, and the formula for cubic equations is so laborious that hardly anyone uses it. In these cases the way forward is to approximate: guess a value, see how far off we are, and correct the guess. The question is how to correct it well. Newton’s example In De analysi, of 1669, Newton chose the equation $$y^3-2y-5=0.$$By trial he saw that the root is near $2$, since $2^3-2\cdot2-5=-1$. So he wrote $y=2+p$, where $p$ is the missing correction, and substituted:

October 11, 2026 Read

The laws of motion: forces and changes in velocity

For Aristotle, a body moves only while something pushes it. Stop pushing the cart and it stops. It seems obvious, and it was accepted for almost two thousand years. Galileo had his doubts. Rolling balls down inclined planes, he noticed that the smoother the surface, the further the ball went. With no friction at all, it would never stop. Descartes reached a similar idea. Newton put these clues together and, in the Principia of 1687, wrote three laws that became the foundation of all mechanics.

October 11, 2026 Read

Gravitation: from the apple to the Moon

If the apple falls, why doesn’t the Moon? Before Newton the answer seemed simple: heaven and Earth followed different rules. On Earth, things fall. In the heavens, the stars and planets turn in perfect circles forever. Newton’s great idea was to distrust that separation: perhaps the Moon is falling all the time. Newton’s cannon The apple tree in the orchard of Woolsthorpe Manor that tradition identifies as “Newton’s tree”.David Smith · Wikimedia Commons · CC BY-SA 2.0 In a draft for the Principia, Newton proposed a thought experiment. Imagine a cannon on top of a very high mountain, firing horizontally. With little powder, the ball lands nearby. With more, it lands further away. But the Earth is round: if the ball is fast enough, the ground curves away beneath it exactly as fast as it falls. The ball falls forever and never reaches the ground. It is in orbit.

October 11, 2026 Read

Light and colours: the prism does not make the rainbow

In Newton’s time it was already known that a glass prism, with sunlight passing through it, produces a coloured band. The accepted explanation came from Aristotle and Descartes: white light is pure, and the glass modifies it, darkening it little by little. A small modification gave red; a large one, blue and violet. Newton had his doubts. What if the colours were already inside white light, and the prism merely separated them?

October 11, 2026 Read

Newton's legacy

Newton was the greatest genius who ever lived, and the most fortunate, for there is only one universe to discover, and it can be discovered only once. — Joseph-Louis Lagrange, as reported by his contemporaries The calculus war Newton created the method of fluxions in 1665–1666 but did not publish it. Leibniz reached calculus independently in 1675 and published in 1684. For some years nobody quarrelled. In 1676 the two even exchanged cordial letters through Henry Oldenburg, and Newton hid his ideas in an anagram.

October 11, 2026 Read

Image credits

Historical images The two historical images come from Wikimedia Commons and are in the public domain. We use the thumbnails provided by Commons, without editing. Portrait of Isaac Newton, Godfrey Kneller, 1689: source page · public domain. Title page of the Principia, 1687: source page · public domain. Blog figures The cover and the prism sketch are original illustrations. The interactive figures for the secant, the approximations of √2 and the orbits were created for this trail. They are teaching aids, not reproductions of manuscripts or simulations of Newton’s experiments.

October 11, 2026 Read

Leibniz against the dots: active and sterile notation

On 29 October 1675, in Paris, Leibniz was adding up infinitely many thin slices under a curve and got tired of writing omnia, “all”, in front of each sum. He noted in his notebook: Utile erit scribi ∫ pro omnia. (It will be useful to write ∫ instead of “all”.) — Leibniz, manuscript of 29 October 1675 The $\int$ is a stretched S, for summa. Two weeks later, in the same notebook, the $d$ appeared, for differentia, to denote infinitely small differences. With these two signs Leibniz had in his hands a language for the whole of calculus. Isaac Newton had reached the same ideas ten years earlier, in Cambridge, but wrote them differently. This is history’s best case for seeing how the shape of a sign changes what can be done with it.

October 13, 2026 Read