Newton's Annus Mirabilis (1665-1666): Calculus, Light and Gravity

Newton’s annus mirabilis, or “year of wonders”, is the name given to the period around 1665–1666 when the young Isaac Newton, in his early twenties, laid the groundwork for calculus, his theory of light and colours, and his ideas about gravity. Much of this work was done at his family farmhouse at Woolsthorpe in Lincolnshire, where he had gone after the Great Plague closed Cambridge University. The episode has become one of the most famous stories in the history of science, though historians stress that it was part of a longer process.

Background: a young scholar at Cambridge

Isaac Newton was born at Woolsthorpe Manor on Christmas Day 1642 by the calendar then used in England (4 January 1643 by the modern calendar). His father had died before he was born, and he was raised largely by his grandmother after his mother remarried. In 1661 he entered Trinity College, Cambridge, as a sizar, a student who paid his way partly by doing tasks for others.

The official curriculum was still based on Aristotle, but Newton read widely on his own. His notebooks show him studying René Descartes, Galileo, the astronomy of Kepler and the mathematics of John Wallis and Frans van Schooten. He took his bachelor’s degree early in 1665.

The plague closes the university

In the summer of 1665 the Great Plague of London spread into the provinces, and Cambridge University closed. Newton returned to Woolsthorpe in about August 1665. He went back to Cambridge in March 1666, left again when plague returned in June, and finally resumed residence in April 1667.

Away from lectures, Newton had time and solitude. Half a century later he recalled that in those years he was “in the prime of my age for invention” and gave more attention to mathematics and philosophy than at any time since. That memorandum, written around 1714 in the middle of his dispute with Leibniz, is the main source for the idea of a single miraculous year.

Mathematics: the birth of fluxions

Over the winter of 1664–1665 Newton found a general form of the binomial theorem, extending it to fractional and negative powers. He then developed what he called the “method of fluxions”, treating curves as traced by moving points and working out their rates of change, or “fluxions”.

By May 1665 he was handling tangents to curves, and by 1666 he could also work backwards from rates of change to areas under curves, recognising that the two problems were inverse to one another. He summed up his results in a manuscript known as the October 1666 tract. This was the core of what we now call differential and integral calculus. Newton did not publish it, which later fuelled the bitter calculus priority dispute with Gottfried Wilhelm Leibniz.

Light and colours

Newton also experimented with glass prisms. The usual view held that white light was pure and that a prism added colour to it. Newton passed a beam of sunlight through a hole in a shutter and a prism and saw that it spread into a long band of colours rather than a round spot.

In his “crucial experiment” he isolated a single colour with one prism and passed it through a second prism. It was bent again but did not change colour. He concluded that white light is a mixture of rays of different colours, each refracted by a different amount. The exact dating of the prism work is debated, and some of it probably took place after 1666, but its foundations belong to these years. He published the theory in the Philosophical Transactions in 1672.

Gravity and the apple

The third strand concerned motion and gravity. Using Kepler’s third law of planetary motion, Newton reasoned that the force keeping the planets in their orbits should weaken with the square of distance. He compared the force needed to keep the Moon in its orbit with the force of gravity at the Earth’s surface and found that they “answer pretty nearly”.

The famous apple enters here. In 1726, the year before his death, Newton told his friend William Stukeley that the idea of gravity had come to him at Woolsthorpe when he saw an apple fall while he sat in a contemplative mood. He told similar versions to his niece’s husband John Conduitt, and Voltaire repeated the story. Newton’s figures did not yet match well, partly because of inaccurate values for the Earth’s size, and he set the problem aside for nearly twenty years.

Life at Woolsthorpe

Woolsthorpe Manor was a modest limestone farmhouse in the hamlet of Woolsthorpe-by-Colsterworth, south of Grantham. Newton’s mother, Hannah Ayscough, had returned there after the death of her second husband, and she had once hoped her son would manage the family farm. As a teenager he had proved a poor farmer, more interested in books and mechanical models than in sheep and crops.

During the plague years Newton seems to have spent most of his time reading, calculating and experimenting. His surviving notebooks are crowded with calculations, sometimes carried to dozens of decimal places, and with notes on mathematics, optics and mechanics. He also experimented on himself: in his studies of vision he pressed a blunt bodkin around the back of his eye to see what coloured circles appeared, and he once stared at the Sun in a mirror for so long that he had to rest in a darkened room for days.

He did not work in complete isolation from Cambridge. He kept in touch with the university, where his mentor Isaac Barrow, the first Lucasian Professor of Mathematics, recognised his ability. Barrow would later help circulate Newton’s mathematical work to other scholars.

Why Newton kept quiet

One striking feature of the annus mirabilis is how little of it Newton made public at the time. He was secretive by temperament and disliked controversy. When he did publish his theory of colours in 1672, it drew criticism from Robert Hooke and others, and Newton reacted badly, threatening to withdraw from scientific exchange altogether. This reluctance meant that his priority for calculus rested on unpublished manuscripts, a fact that became important decades later.

Myths and facts

• Myth: the apple hit Newton on the head. No early account says so; the bump on the head is a later embellishment.
• Myth: Newton worked out universal gravitation in 1666. He had key insights, but the complete theory of universal gravitation came in the 1680s and was published in the Principia in 1687.
• Myth: everything happened in one year. Newton’s own notes show work spread from 1664 to 1667, and historians such as Richard Westfall have warned against treating the period as a single flash of genius.

Consequences

When Newton returned to Cambridge he was elected a fellow of Trinity College in 1667. In 1669 Isaac Barrow resigned the Lucasian chair of mathematics, and Newton, aged 26, succeeded him. His reflecting telescope, shown to the Royal Society in 1671–1672, earned him election as a Fellow in January 1672.

The plague-year ideas fed directly into his later masterpieces: the Principia (1687) and the Opticks (1704). They also shaped mathematics and physics for the next two centuries.

Timeline

• 1642, 25 December (Old Style): Isaac Newton born at Woolsthorpe Manor.
• 1661: Newton enters Trinity College, Cambridge.
• 1665, early: Newton receives his BA; works on the binomial theorem.
• 1665, summer: Plague closes the university; Newton returns to Woolsthorpe.
• 1666, March–June: Newton briefly back in Cambridge before leaving again.
• 1666, October: Newton writes his tract on fluxions.
• 1667, April: Newton returns to Cambridge; elected fellow of Trinity in October.
• 1672: Newton’s theory of light and colours published.

Why it matters

The plague years show how a period of isolation allowed a gifted young scholar to bring together the mathematics, optics and astronomy of his time in new ways. Calculus, the theory of colours and the idea of universal gravitation all trace their roots to Newton’s notebooks of 1665–1666, even though they took decades to mature.