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      <text>10.4.2 (History of calculus)</text>
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      <text><![CDATA[<h3>History of the Calculus</h3>

<p>Calculus as an entity is perhaps one of the most modern branches of mathematics that we have discussed in this course. However, as we have seen calculus is related to the ideas of infinite series and sums, so we can trace the earliest seeds of calculus back to ancient times, with the ancient Greeks. Recall that questions about infinite series and limits were raised by Zeno in around 450 BCE. This led to the method of exhaustion with was then employed and expanded by Archimedes in around 225BCE who used the method of exhaustion to find the area of certain parabolas and then he gave an approximation to the area of a circle. Remember that in some sense calculus is involved with trying to find the area of complicated shapes, mapped out by curves, by using the notion of infinite Riemann sums. These ideas came a lot later, which makes Archimedes results even more remarkable. In fact, in Europe at least, Archimedes' method stood until the 17th century. </p>
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<p>One of the earliest examples of an integration can be traced back to the medieval Arabic empire and the mathematician Ibn al-Haytham who in the 11th century who gave a formula for the sum of fourth powers. He used the methods to find the volume of a paraboloid using what is loosely an integration argument. In the 12th century Persian mathematicians appear to have given the derivative of a cubic. At around the same time in history, some ideas on calculus came out of the famous Kerala school of Astronomy and Mathematics. Whilst these Indian mathematicians did not discover the ideas of the integral or the derivative, they did manage to state some infinite series approximations to certain functions. The major breakthrough in calculus would have to wait hundreds of years, until 17th century European mathematicians turned their attention to the problem.</p>
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<p>The story of calculus is the story of Newton and Leibniz. This is perhaps the biggest, most public and most bitter rivalry in mathematics as both claimed to be the originator of differential calculus. Prior to the work of these two mathematicians, several European scholars had begun to study topics which we would now label as calculus. Mathematicians including Barrow, Fermat, Descartes, Pascal and Wallis discussed the idea of a derivative. In fact Fermat went further and developed a method of finding tangents and maxima and minima of different curves which was related to modern differentiation. In Newton's later work he would write that his work on derivatives was inspired by Fermat's work. In the 1630s and 40s the mathematician Cavalieri developed a method called the method of indivisibles as a way to perform integration, this was a more modern version of the method of exhaustion. However, he was not rigorous about his work and it is not clear today how he viewed his work.  Barrow later considered the problem of motion with variable speed. We will see that the derivative of the distance is velocity and the inverse operation takes one from the velocity to the distance. Hence an awareness of the inverse of differentiation began to evolve naturally and the idea that integral and derivative were inverses to each other were familiar to Barrow. In fact, although he never proved it he was working towards the Fundamental Theorem of Calculus.</p>
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<p>It sounds as though much of calculus was built before Newton and Leibniz were even born. Actually, this is not quite true, so far we have mentioned only disparate ideas that were developed by various mathematicians. The entirety of the theory of calculus wouldn't come together until Newton and Leibniz. In 1666 Newton wrote a theory on 'fluxions' which was not published, but is thought to have heavily influenced the direction calculus would take. Newton would introduce the gradient of the tangent and the concepts of antiderivatives. Newton came to calculus from his work in physics and geometry, he viewed the field as the mathematical description of changes in motion. As Newton would not circulate this work until much later in the 18th century, at the same time, separately Leibniz was developing his own version of calculus. Where Newton viewed calculus as the change in given variables over time, whereas Leibniz thought of variables ranging over sequences of infinitely close values called infinitesimals. Leibniz was keenly aware of the requirement of a good notation and would introduce the notation \(\frac{dy}{dx}\). Newton however often wrote for his own benefit and was liable to alter notation to suit himself. It is for this reason that most of what we recognise as calculus today comes from the version Leibniz developed and published in 1686. This was after Newton's study of fluxions but before he published it. This led to a great rivalry between the two men, and indeed between the English and German scientific communities. Who was the first to invent calculus? It probably depends on your point of view. </p>
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<p>Whilst calculus originated with newton and Leibniz, advanced didn't stop with them. Calculus had been created but there was a general lack of rigour in the theory and logic underpinning it. Mathematicians therefore sought to place calculus on a rigorous basis. This culminated in the work of Cauchy in the 19th century and the development of mathematical analysis.</p>]]></text>
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