From the seventeenth century, Europe replaced the Middle East as the engine house of mathematical ideas. Du Sautoy visits Urbino to introduce Perspective using mathematician and artist, Piero della Francesca's The Flagellation of Christ.

Du Sautoy proceeds to describes René Descartes realisation that it was possible to describe curved lines as equations and thus link algebra and geometry. He talks with Henk Bos about Descartes. He shows how one of Pierre de Fermat’s theorems is now the basis for the codes that protect credit card transactions on the internet. He describes Isaac Newton’s development of math and physics crucial to understanding the behaviour of moving objects in engineering. He covers the Leibniz and Newton calculus controversy and the Bernoulli family. He further covers Leonard Euler, the father of topology, and Gauss' invention of a new way of handling equations, modular arithmetic. He mentions János Bolyai.

The further contribution of Gauss to our understanding of how prime numbers are distributed is covered thus providing the platform for Bernhard Riemann's theories on prime numbers. In addition Riemann worked on the properties of objects, which he saw as manifolds that could exist in multi-dimensional space.

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