Calculus Takes Form

In the 16th Century, the order of the day was mechanics. As such it suddenly seemed useful to consider once again the concepts of rates of change and area. Other motivations for continuing this research lay in astronomy – planetary motion was a favourite among these topics – and a number of more ‘pure’ concepts.

The mathematicians who made contributions to this work are numerous, with major work from Fermat, Roberval and Cavalieri laying the scene for the entry of the greats of calculus. The two men who are acknowledged as the fathers of modern calculus are Sir Isaac Newton and his contemporary from across the channel, Gottfried Leibniz. These two mathematicians independently reached the same conclusions, but neither man was able to sufficiently establish the logical roots of their work. This was despite the fact that both men had methods that could equally well answer complicated questions, reaching the same, correct answer every time.

The fact that these two men independently reached the same results was a point of contention for many years. Newton in particular was sure than Leibniz had based his work upon Newton’s, and remained bitter towards the German mathematician for the rest of his life. When Newton was appointed to a position in London’s Royal Society, he used his position to influence the argument, to the point where he personally chaired the ‘independent’ board that investigated the matter, and published the findings under another name.

The methods used by both men were frequently attacked by other, disbelieving mathematicians and even theologians, for in particular the use of ‘infinitesimals’ was not seen as a mathematically sound method. It was not until the 19th Century when Cauchy published his definition of what we today call the ‘limit’ that any formal and rigorous basis was established for the calculus.