the Discourse had influenced the work of mathematicians and astronomers throughout Europe. Such men as the English mathematicians William Wallace and Isaac Barrow, as well as natural philosophers and mathematicians on the Continent, led by Pierre de Fermat and Christiaan Huygens, used Descartes’s findings as a springboard for their own efforts, which began to focus on the properties of the curves that could be drawn using Cartesian coordinates.
A simple example is the graph produced by plotting the distance travelled by a ball dropped from a high tower against the time for which it has fallen. Galileo had shown that the speed of a ball increases with time. If after one second the ball has fallen 16 feet, after two seconds 64 feet and after three seconds 144 feet, clearly it is accelerating. If these values are plotted on a graph with speed on the y-axis and time on the x-axis a curve is produced.
Figure 1. The curve produced by plotting distance against time.
Now it is comparatively easy to calculate properties for straight-line graphs. For example, the area under a straight line can be calculated by simple geometry known to the Babylonians, and the gradient of a straight line (or its steepness) can be found by dividing the change in the values along the y-axis by the corresponding values along the x-axis. So, if the distance-time graph had been a straight line, the gradient would have given us the speed of the ball (the change in
Figure 2. Calculating the gradient and the area under a straight line.
distance with time). But how can the properties of a curve be calculated?
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.
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 HEISENBERG1
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.2
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’.3 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.’4
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
