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Watch The Story of Maths online: Episode 3 The Frontiers of Space

By the seventeenth century Europe had taken over from the Middle East as the worlds power house of mathematical ideas. Great strides had been made in understanding the geometry of objects fixed in time and space. The race was now on to discover the mathematics that describes objects in motion. In this programme, Marcus du Sautoy visits France to look at the work of Ren Descartes, an outstanding mathematician and theoretical physicist as well as one of the great philosophers, who realised that it was possible to link algebra and geometry. His vital insight - that it was possible for curved lines to be described as equations - would change the course of the discipline forever. Marcus also examines the amazing properties of prime numbers discovered by Pierre Fermat, whose famous Last Theorem would puzzle mathematicians for more than 350 years. He shows how one of Fermats theorems is now the basis for the codes that protect credit card transactions on the internet. In England he looks at Isaac Newtons development of calculus, a great breakthrough which is crucial to understanding the behaviour of moving objects and is used today by every engineer. He also goes in search of mathematical greats such as Leonard Euler, the father of topology or bendy geometry and Carl Friedrich Gauss, who at the age of 24 was responsible for inventing modular arithmetic (a new way of handling equations). Gauss made major breakthroughs in our understanding of how prime numbers are distributed. This made a crucial contribution to the work of Bernhard Riemann, who developed important theories on prime numbers and had important insights into the properties of objects, which he saw as manifolds that could exist in multi-dimensional space.

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