Isaac Newton: The Last Sorcerer

It was soon realised that one way to determine properties of curves, such as the one in our problem, was to imagine them as constantly shifting straight lines: if a straight line was drawn next to a curve and touched it at a particular point, this line could approximate the curve at that point. Mathematicians called this straight line a tangent, and found that they could treat a tangent like any other straight line – they could, for example, find its gradient and therefore work out a value for the speed of the ball at that particular point. But this was still an approximation – and a very limited one at that.

Simple problems concerning objects travelling in circular motion had been studied by earlier generations of philosophers, especially Galileo, but by the 1660s astronomers weaned on Kepler’s work were becoming interested in mathematical models to describe the

Figure 3. The tangent to the curve.

new celestial mechanics – the mathematics of how the planets maintain their orbits around the Sun. They of course realised that the mathematics of curves could lead to a fuller understanding of planetary motion, but limited solutions such as those offered by drawing tangents were not accurate enough to correlate with increasingly sophisticated methods of gathering observational data. Although the mathematicians and astronomers of Europe were exploring methods of working with curves and some, such as Fermat and the great English polymath Christopher Wren, came to very limited solutions that worked in specific cases, there was a need for general solutions, or methods that could be applied to all situations. Newton gradually became aware of this as he studied the work of his predecessors while an undergraduate student at Cambridge during the early 1660s. By the middle of the decade all the elements were in place for a mathematician of genius to produce the required new mathematics. And, thanks to a series of unpredictable events, Newton was able to find the time and inspiration to do just that.

Chapter 5 A Toe in the Water (#ulink_da3255b7-3d60-57a7-8870-9987efae0c33)

It is probably true quite generally that in the history of human thinking the most fruitful developments frequently take place at those points where two different lines of thought meet. These lines may have roots in quite different parts of human culture, in different times or different cultural environments or different religious traditions: hence if they actually meet, that is, if they are at least so much related to each other that a real interaction can take place, then one may hope that new and interesting developments may follow.

WERNER HEISENBERG

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When, in the spring of 1669, the Trinity fellow Francis Aston was preparing to leave on a European tour, he wrote to his friend Isaac Newton asking for his advice on how best to conduct himself and what to look out for on his travels. This is surprising, since Newton had never travelled abroad and had only recently made his first trip to London. But it illustrates the high esteem in which Newton was held by his colleagues so early in his career, even in connection with matters outside his area of expertise. More significant still is Newton’s reply to Aston’s letter, for, as well as asking his friend to gather alchemical information for him and to attempt to track down the famous alchemist Giuseppe Francesco Borri, then living in Holland, Newton went on to offer a long list of dos and don’ts as though he were a seasoned globe-trotter. These included the recommendation:

If you be affronted, it is better in a foreign country to pass it by in silence or with jest though with some dishonour than to endeavour revenge; for in the first case your credit is none the worse when you return into England or come into other company that have not heard of the quarrel, but in the second case you may bear the marks of the quarrel while you live, if you outlive it at all.

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The reason for this easy confidence is that by the 1660s Newton had already adopted what one of his biographers has called ‘a Polonius-like pose’.

(#litres_trial_promo) Even as a boy he had been confident to the point of alienating others, but Newton the man, the twenty-six-year-old fellow of Trinity College, Cambridge, six months away from accepting the Lucasian chair, was already so accomplished that if he had done nothing further with his life he would still have found a significant place in the history of science.

Although his genius was realised by only a handful of associates in Cambridge and he was totally unknown to the scientific community, by 1669 Isaac Newton was in fact the most advanced mathematician of his age, creator of the calculus as well as elucidator of the basic principle behind the inverse-square nature of gravity and the theory of the nature of colours. Within the space of four years he had grown from unnoticed undergraduate to a man on the foothills of greatness. But, while he had been internally fostering these scientific upheavals, catastrophes had befallen the larger, external, world – catastrophes that had even threatened the ivory tower that Newton inhabited at the very heart of academe.

The plague of 1665 was not the first in English history, but coming as it did straight after the Civil War, and taking the lives of almost 100,000 people (some 70,000 of them in London, which then had a population of under half a million), it was seen by many as yet another fulfilment of the prophecies in the Book of Revelation. The fact that it extended into the year 1666, with its numeric similarity to the ‘sign of the beast’, only made the psychological impact of the catastrophe more poignant. Daniel Defoe reports that ‘Some heard voices warning them to be gone, for that there would be such a plague in London, so that the living would not be able to bury the dead.’

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Some 300 years earlier the Black Death had killed an estimated 75 million people in Europe – about a third of the population – but, because most people of the seventeenth century could neither read nor write, it is unlikely that any but the educated few would have realised that plague was a relatively common occurrence. Their only likely knowledge of the virulence of such diseases would have come from their grandparents and great-grandparents recounting horror stories of the last major outbreak, forty years earlier, in 1625.

The plague began in London and spread to other parts of the country rapidly during the hot summer of 1665. It was always at its worst in the east of the city, in the districts of Stepney, Shoreditch, Whitechapel and around the crowded streets clustered at the foot of St Paul’s. At its height it claimed 10,000 lives a week, and in one day in September 1665 alone, 7,000 victims died. The disease was in fact bubonic plague – a bacterial infection carried by a flea which infested the black rat (Rattus rattus). Wherever rats could breed, the disease spread like wildfire. The flea carried the initial infection to humans via a bite. There was no cure and only a slim chance of survival for those unfortunate enough to become infected. Without the benefit of antibiotics, the only means of containing the disease was quarantine.

By the end of the first summer of the Great Plague, after tens of thousands had lost their lives, the quarantine laws which would eventually help to halt the spread of the disease were finally enacted and major towns and cities became citadels where travellers and visitors were entirely unwelcome. It is clear from a number of reports of the spread of the disease that it took some time for the authorities to realise they were facing a major catastrophe, and by the time they did the plague had a grip on London and had been carried to many other parts of the country. Samuel Pepys, the great monitor of the Zeitgeist, first mentioned the plague in his diary entry of 30 April 1665, noting ‘Great fears of the sickness here in the City, it being said that two or three houses are already shut up.’

(#litres_trial_promo) But it was not until 15 June that he reported, ‘The town grows very sickly, and the people to be afeared of it – there dying this week of the plague 112, from 43 the week before.’

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Cambridge escaped relatively lightly, and the university fared amazingly well. The first mention of the disease appears in the Annals of Cambridge of August 1665, when, we are told, the plague had prevailed and taken the life of one of the town bailiffs, a William Jennings. Things grew worse as the summer progressed, and the Stourbridge fair was cancelled that year (and in 1666) because of the fear of attracting travellers to Cambridge – especially those from the capital. According to the Annals, only 413 people died in all the parishes of Cambridge during 1665, and many of these deaths were from natural causes. They then go on to report that during a two-week period in November of that year a total of fifteen deaths from plague were recorded.

(#litres_trial_promo) In the colleges, there was not one case of the disease all year, largely because the majority of students, fellows and staff had left during the early summer, and those few who did stay kept any contact with the townsfolk to an absolute minimum, locking themselves away in their sanctuaries like medieval monks.

The exact date when Newton left Cambridge is unclear. He was certainly there on 23 May, because he paid his tutor Pulleyn £5.

(#litres_trial_promo) He was not in college for most of July and early August (the college was dismissed on 8 August), because he did not claim six and a half weeks worth of commons (food allowance) paid to those who had stayed on to risk plague during the summer. According to most accounts, he left Trinity around the end of June or the beginning of July and did not return, except for a brief spell in early 1666, for almost two years.

He travelled to his mother’s home, the manor in Woolsthorpe, where tradition has it that he made his great discoveries concerning gravity and the mathematical breakthroughs that later made him famous. It is in Woolsthorpe, in the orchard next to the house, that the famous apple is supposed to have dislodged itself with impeccable timing and set in motion the development of the theory of gravity. Thereafter, one might assume, the Principia was a mere formalising of the great revealed truth. Yet the reality, magnificent though it was in its intellectual depth and its effect upon the course of science, was far more prosaic. The truth is not so much grounded in singular fluke events or any deeply symbolic psychological drives associated with Newton living in the home of his childhood than it is to do with a gradual revelation brought about by concentration and sheer dedication. As Newton himself said when asked how he came upon his great discoveries, ‘I keep the subject constantly before me, till the first dawnings open slowly, by little and little, into the full and clear light.’

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The apple story is almost certainly a fabrication, or at the very least a highly embroidered version of the truth. Indeed, the very notion, so integral to many early accounts of Newton’s life, that there were two special years in his life during which everything was solved – the so-called anni mirabiles of 1665 and 1666 – is an extreme simplification of the facts. Although Newton’s achievements during the time he spent in Woolsthorpe sprang from intuition and inspiration and did lead to the great laws that lie as a foundation beneath our technology, they did not appear fully formed and complete. Although the years 1665 and 1666 were truly great ones for Newton’s intellectual development, they mark merely the start of his quest. If we are to label Newton’s achievements by the calendar, then the true anni mirabiles cover more than two decades, from his arrival in Woolsthorpe to the delivery of the Principia in 1687, and encapsulate his period of almost single-minded dedication to the practice of alchemy during the 1670s and ’80s as well as the gradual transmutation of his intuitive insights into hard science.

Quite how and indeed from where the initial moment of inspiration came remains a mystery, and, despite the anecdotes and varied accounts describing Newton’s efforts during 1665 and 1666, we may never know how one of the most important sets of scientific and mathematical discoveries in history was initiated.

The story of the apple has come down to us from a number of sources. First there is William Stukeley. During the spring of 1726, a year before Newton’s death, the biographer visited the great scientist in his final home in Kensington. As they walked out into the garden of the house, Newton remarked that it had been on just such an occasion that he had first realised the theory of gravity. Intrigued, Stukeley pursued the matter ‘under the shade of some appletrees, only he and myself,’ Stukeley recounts. ‘Amidst other discourse, he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. It was occasioned by the fall of an apple, as he sat in the contemplative mood.’

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Another account comes from Newton’s great admirer Voltaire, who made the English scientist famous in France with his Éléments de la philosophie de Newton (1736), in the English edition of which he says:

One day in the year 1666, Newton, having returned to the country and seeing the fruits of a tree fall, fell, according to what his niece, Mrs Conduitt, has told me, into a deep meditation about the cause that thus attracts bodies in the line which, if produced, would pass nearly through the centre of the Earth.

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As the passage relates, Voltaire received this story second-hand from Newton’s half-niece Catherine Barton, wife of John Conduitt. Voltaire never met Newton.

There is one other contemporary account of note: that of Henry Pemberton, who was the editor of the third edition of the Principia, published in 1726. He describes the scene in a similar way: ‘The first thoughts, which give rise to his Principia, he had, when he returned from Cambridge in 1666 on account of the plague. As he sat alone in the garden, he fell into a speculation on the power of gravity.’

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The common factor in all these versions of the story is that they derive directly from Newton himself and we therefore have only his word that the story is true. Perhaps, one day during the summer of 1666, he did sit under a tree and see an apple fall and it was this, combined with a wealth of other factors, that inspired his theory. But it is also quite likely that the apple story was a later fabrication, or at least an exaggeration designed for a specific purpose – almost certainly to suppress the fact that much of the inspiration for the theory of gravity came from his subsequent alchemical work.

On a prosaic level, Newton’s work as an alchemist was completely anathema to the traditional world of science and to society in general. But, beyond that, attempting to transmute base metals into gold – a preoccupation of the alchemists – was a capital offence. Even in old age Newton was determined to maintain his duplicity, both to protect himself and to preserve unsullied his hard-won image as the greatest scientist who had ever lived.

So, if these stories are false, how then did Newton arrive at the inverse square law for gravity – the first major development towards the elucidation of universal gravitation, the principle that all masses attract all other masses?

The first step was to use his mathematical studies in order to mould a set of general mathematical principles that he could use to investigate planetary motion. Since his earliest inquiries into basic mathematics, begun two years earlier, during the spring of 1664, he had managed to assimilate the entire canon of known mathematics and then to extend it into totally uncharted waters – ‘For in those days I was in the prime of my age of invention,’ Newton said of himself sixty years later.

(#litres_trial_promo) He was familiar with the latest work on the mathematics of curves and the principle that tangents can approximate to the curve and allow certain calculations to be managed, but, like many mathematicians of the time, he wanted something more precise. In particular, he was interested in finding the area under a curve (the area between a curve and the x-axis) and a precise value for the curvature (or gradient) of a curve.

Scholars are in general agreement that the greatest influence upon Newton’s own thinking about these problems came from his reading Descartes’s Geometry during the summer of 1664, but others have pointed out that Isaac Barrow had also made some considerable progress with the mathematics of gradients and curves and that Newton may have learned a great deal from both men.

During 1665 and early 1666 Newton worked on these problems and devised a method of finding the exact gradient of a curve by a method which has since become known as differentiation. To understand this method we must first recall that a graph is a way of representing a set of values describing a situation. In the last chapter, the example of a ball falling from the tower was used to illustrate how a real situation can be described graphically. Equally, an algebraic equation is another way of describing a situation. In fact a graphical representation and an algebraic description can both represent the same thing. This means that the algebraic and graphical representations are paired, so manipulating equations by a suitable method can lead the mathematician to information about the curves these equations mirror.

Newton’s greatest mathematical breakthrough was the realisation that a particular manipulation of a suitable equation could lead to a precise value for the gradient of the curve represented by that equation. This method of manipulation is the essence of differentiation. Another process carried out on a equation (a process since named integration) leads the mathematician to the area under a curve represented by that equation. The calculus is the overall term for these two processes of differentiation and integration, and together they are powerful tools for the mathematician and the scientist.

Although sometimes placed in his ‘Woolsthorpe period’, work on this development was actually begun while Newton was still in Cambridge. By his own account, he had begun to develop the calculus as early as February 1665.

(#litres_trial_promo) His first mathematics paper, dealing with a mathematical process called the summation of infinitesimal arcs of curves (a major step towards a full realisation of the techniques involved in the calculus), was composed in May 1665.

Once he had a general method for the calculus, the next step was to apply it to the practical problem of planetary motion – how planets orbit the Sun, and the Moon the Earth, and how mathematical laws can represent these movements.

A thought experiment familiar to natural philosophers was that of the stone on a string. This may be visualised by imagining a stone attached to a string being whirled around the experimenter’s head. In this model, one force draws the stone towards the centre of the circular path and another pulls the stone away. The Dutchman Christiaan Huygens called the first of these forces the centripetal force and the other the centrifugal force. The stone continues to travel in a circle around the experimenter’s head because the two forces cancel out. If the string is cut the stone will fly off in a straight line at a tangent to the circle.

Using this as a basis, Newton created a thought experiment to determine a way of calculating the outward or centrifugal force experienced by an object travelling in circular motion. To begin with, he imagined a ball travelling along the four sides of a square inscribed in a circle.