In the third episode we will see Europe by the 17th century taking over from the Middle East as the powerhouse of mathematical ideas. Great strides had been made in understanding the geometry of objects fixed in time and space. The race was on to discover the mathematics to describe objects in motion. This programme explores the work of Rene Descartes, Pierre Fermat, Isaac Newton, Leonard Euler and Carl Friedrich Gauss. 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 J. M. 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 Leonhard Euler, the father of topology, and Gauss' invention of a new way of handling equations, modular arithmetic. 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.

Two thousand years ago one of history's most notorious individuals was born. Professor Mary Beard embarks on an investigative journey to explore the life and times of Gaius Julius Caesar Augustus Germanicus - better known to us as Caligula. Caligula has now become known as Rome's most capricious tyrant, and the stories told about him are some of the most extraordinary told about any Roman emperor. He was said to have made his horse a consul, proclaimed himself a living God, and indulged in scandalous orgies - even with his own three sisters - and that's before you mention building vast bridges across land and sea, prostituting senators' wives and killing half the Roman elite seemingly on a whim. All that in just four short years in power before a violent and speedy assassination in a back alley of his own palace at just 28-years-old." But how much of his story is true? Travelling across the Roman world - from Germany and Capri in the bay of Naples to the astonishing luxury of his life in imperial Rome - Mary attempts to peel away the myths. Some stories are difficult to get to the bottom of as they were written by hacks long after his death, but there is plenty of surviving evidence where the 'real' Caligula can be glimpsed. Such as in the extraordinary luxury of his private yachts outside Rome; in the designs he chose for his coins when he became Emperor; in an eye-witness account of Caligula's withering humour written in 41AD; the trial documents covering the mysterious death of his father when he was just seven-years-old; and even in a record of his imperial slaves - from the palace spy to his personal trainer. Piecing together the evidence, Mary puts Caligula back into the context of his times to reveal an astonishing story of murder, intrigue and dynastic family power. Above all, she explains why Caligula has ended up with such a seemingly unredeemable reputation. In the process, she reveals a more intriguing portrait of not just the monster, but the man.

Sagan discusses the story of the Heike crab and artificial selection of crabs resembling samurai warriors, as an opening into a larger discussion of evolution through natural selection (and the pitfalls of intelligent design). Among the topics are the development of life on the Cosmic Calendar and the Cambrian explosion; the function of DNA in growth; genetic replication, repairs, and mutation; the common biochemistry of terrestrial organisms; the creation of the molecules of life in the Miller-Urey experiment; and speculation on alien life (such as life in Jupiter's clouds). In the Cosmos Update ten years later, Sagan remarks on RNA also controlling chemical reactions and reproducing itself and the different roles of comets (potentially carrying organic molecules or causing the Cretaceous–Paleogene extinction event).

With optional Hebrew subtitles. Simon Schama explores the history of the Jewish experience from ancient times to the present day. This programme begins the tale 3,000 years ago, with the emergence of a tribal people in a contested land and their extraordinary book, the Hebrew Bible, a chronicle of their stormy relationship with a faceless, formless, jealous God. It was loyalty to this 'God of Words' that defined the distinct identity of the ancient Jews and preserved it despite all that history could throw their way". The story unfolds with a dazzling cast of historical characters: Sigmund Freud dying in exile in London; Victorian evangelicals and explorers following 'in the footsteps' of Moses; Jewish mercenaries living, prospering and intermarrying in the pagan land of Egypt; Messianic Jews dreaming of the Apocalypse; and a Jewish historian, Josephus, who witnessed first-hand the moment when the apocalypse finally came and the Romans destroyed the Jewish High Temple in Jerusalem.

Saturn's secrets are out. The ferocious weather, the evolving ring system and the discovery of active geology on Saturn's moons has rewritten the textbooks. Scientists are looking for life on Saturn's moons and they may have found it on Titan.

The digital era gives rise to unprecedented opportunity to exploit, and be exploited. Troves of child pornography are available at the click of a mouse. Who are these children - and who profits from their abuse? While facilitating the darker side of our impulses, technology is also being touted as a cure to rehabilitate predators. In this episode meet the children who are forced to perform sex acts on camera; people who use the same technology to protect kids and nab the perpetrators; plus, a pedophile who reveals the roots of his addiction.