The Calculus Wars

Review by Andrew Crumey

Published in Scotland on Sunday

The Calculus Wars
Jason Bardi
High Stakes, £16.99

One of the most famous spats in scientific history was the war of words between Isaac Newton and the German philosopher-scientist Gottfried Leibniz over which of them had been first to discover the mathematical theory of calculus.

Jason Bardi, a biologist by training, explains the scenario in the introduction to this, his first book. Calculus, he explains, is “useful for investigating everything from geometrical shapes to the orbits of planets in motion around the sun”. Newton was technically the first to discover it, but failed to publish his results. Leibniz subsequently went into print, and was accused of having stolen Newton’s ideas. He was no cheat, however, and it now seems that both men arrived at the same results independently.

It has the potential to be a fascinating tale, but Bardi’s opening synopsis lays bare the problems he faces, and is unfortunately unable to overcome. First there is the fact that we know from the outset how the story ends, with the posthumous declaration of an honourable draw. So we cannot expect much in the way of suspense. Secondly, and perhaps more seriously, there is the problem of understanding exactly what it was that Newton and Leibniz were fighting over. What is calculus? Again, though, Bardi goes little further than the bare sketch he gives in the introduction.

Calculus began as a way of understanding speed. Can we speak of the speed of a car, say, at a particular moment in time? In a single instant, a car travels no distance – yet Newton and Leibniz both found how to define instantaneous speed – the thing we now measure with a speedometer. This is “differential” calculus. If you draw a graph of the car’s speed as it varies over time, you get a wavy curve, and the area under the curve is the distance the car has travelled – this is called “integration”.

It only takes a few lines to explain, but that is more than Bardi can spare in his tale of intellectual warfare. And since the battle was fought entirely by letters, nearly all of them written by intermediaries supporting one or other side, the tale is not exactly riveting.

Still, there are a few nuggets here that will appeal to maths teachers looking to pad their lessons with historical background. I was interested, for example, to note that the first ever book on calculus to be published in Britain was by a Scotsman called John Craig, “something of a forgotten player in the invention of calculus”. People who like to think that the Scots invented everything can therefore throw a new spanner in the works by proposing Craig as the third father of calculus. Craig appears to have got his ideas from Leibniz, since he used all the same terminology, but he was still credited for a while – in Germany at least – as having beaten Newton.

It was moreover another Scot, John Keill, who set off the whole spat by accusing Leibniz of plagiarism. There followed years of international arguing which ended only with Leibniz’s death from gout, leaving the elderly Newton satisfied but still indignant. If Bardi fails to make these larger-than-life characters come to life, his supporting cast may nevertheless offer sufficient interest to readers already well enough acquainted with the ideas they fought over.