The development of calculus – or the measurement of rates of change, if you’ve never really understood what calculus was but just nodded wisely when your teacher mentioned it – was not only one of the most significant advances ever in mathematics, but also led to one of the nastiest disputes in the history of science. It took place over nearly fifty years from the late 1660s to the first couple of decades of the eighteenth century and involved two of the greatest thinkers of all time: Gottfried Leibniz and Isaac Newton.
Before we explore calculus, it might be interesting to take a look at the lives of these two great physicist-mathematicians. We’ll start with Newton, as he’s better known in Britain than his German counterpart. He came up with the theory of gravity when an apple fell on his head – or so he said, although he was not always very trustworthy, especially when talking about his own achievements.
Newton had a history of arguing with his contemporaries. In the first place, Robert Hooke accused Newton of stealing his ideas on gravity. Now, Hooke was no lightweight. He was a great physicist who discovered the law of elasticity, found that Jupiter rotated on its own axis, first used the biological term ‘cell’, researched into fossils, helped to re-build London after the Great Fire there in 1666, wrote a remarkable book on how the memory works, and many, many other things.
It’s odd, then, that we have no picture of him and he was so little-known until quite recently. Why, we could ask, was he so completely forgotten? In fact, it was Newton who destroyed Hooke’s reputation and probably had the only image of him burnt after his death.
Then, there was the strange case of Edmond Halley – the man that the comet is named after and who did important research in astronomy and also made one of the first demographic studies of Britain. Halley went to ask Newton about Kepler’s law of the movement of the planets. Newton claimed that he had already solved the problem, but could not find his notes. Many think he worked on it afterwards and later published his findings without mentioning that Halley focussed his interest on the subject.
Those are only two examples of Newton’s rather difficult personality. But it’s worth remembering that there are very few stories about the great scientist that are reliable. There always seem to be different versions of events.
Gottfried Leibniz was a man of many talents too. He is not only famous for his development of calculus and making the first calculator which could multiply and divide as well as add and subtract, but also a major philosopher. Apart from his better-known work, he was interested in rescuing the German-speaking people from the terrible uncertainty following The Thirty Years War, which ended around the time of his birth and which killed a third of all the Germans.
He tried hard to persuade the French king, at that time the most powerful in Europe, not to attack Germany but to direct his attention to Egypt instead. He was unsuccessful. But, not discouraged, he went to London to try again there. While Leibniz was in England, he met many scientists. However, this journey was to come back to haunt him.
In 1684, Leibniz published his classic work on calculus. By the end of the decade, he was the principal mathematician in Europe. Twenty years later, a follower of Newton’s complained that Leibniz’s work was, in fact, really Newton’s. Nowadays, ownership of intellectual property is shown by patent and publication, but it was not always so easy. In the first place, there were no patents and, in the second, research papers and even books could take many years to travel from where they were written to the attention of scientists in other lands.
If it was a question of first publication though, then Leibniz was clearly first. The difficulty was that Newton suggested that someone had shown Leibniz research that the English scientist had already done while the German was visiting England twenty or more years before and that Leibniz had used this without mentioning that his own work was based on it.
Leibniz was furious. He wrote a letter in 1708 demanding an apology. Instead the Royal Society set up a committee in 1712 to investigate. In the meantime, the two physicists exchanged angry letters. Even in the seventeenth century when the post was slow, twenty years was a very long time to wait to make a complaint, as Leibniz pointed out. Newton showed though that his work from the late 1660s and 1670s had already made great progress in the invention of calculus long before Leibniz ever picked up a maths book.
The difficulty was made more complicated by differences in the approaches of the two men: Newton, like Leibniz, had studied classics – there was no other choice in those days, as mathematics and sciences were not offered at universities – and continued the Ancient Greeks’ interest in geometry, the mathematics of shapes. Leibniz, by contrast, was fascinated by the Arab invention of algebra and used this in his calculus. (One of his many accomplishments was fluency in Arabic.)
This European-British divide over the advantages of algebra and geometry remained until the early nineteenth century, when British scientists at last adopted Leibniz’s approach. However, the compromise that this difference of approach offered – that the two men had arrived at the same point by different geometrical and algebraic routes – was wasted, as scientists supported one scientist or the other, based on their nationality.
The differences became more philosophical as the battle between the two men continued. To Newton, the fundamental measurement of everything was time. Calculus was the science of how things change with time. Not so for Leibniz. He was far more interested in how things change in relation to each other and, so, the organisation of space. As Leibniz explained in yet another angry letter, a calculus that depended on time meant that God had created a world that needed constant intervention. Why was Newton’s God such a “sloppy watchmaker” that he needed to re-visit his creation to keep refining it? To Leibniz, a man who was just as religious as Newton was, God had created a perfect universe that did not need him to keep interfering with it to get it absolutely right.
None of this was helped by the language that the two men used to each other in their letters. In one, for instance, Leibniz calls Newton’s supporters “apes”.
Meanwhile, in London, the Royal Society was continuing its investigations. Unfortunately, it was Newton who was writing the Society’s response. It was a case of Newton judging Newton. He pointed out that Leibniz had a habit of going back over his journals and revising them by including later discoveries. How could the world then trust what he said he had written thirty years before? In other words, Leibniz was a liar. Newton and the Royal Society, of course, found that the German had stolen his ideas.
For Leibniz, the argument meant that his employer, the ruler of Hanover, never invited him from the small German city-state to Britain, when he became King of England in 1714. He had to continue his work without the company of the greatest scientific thinkers, who were mostly working in Paris and London. Newton had actively plotted against Leibniz.
Still, it was Leibniz’ version of the calculus that was universally used from the nineteenth century. Had he borrowed his ideas from Newton? We will probably never know, but, most likely, the two men arrived at their algebraic and geometrical proofs independently.