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.

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.

Andrew Marr tells the story of the first empires which laid the foundations for the modern world. From the Assyrians to Alexander the Great, conquerors rampaged across the Middle East and vicious wars were fought all the way from China to the Mediterranean. But this time of chaos and destruction also brought enormous progress and inspired human development. In the Middle East, the Phoenicians invented the alphabet, and one of the most powerful ideas in world history emerged: the belief in just one God. In India, the Buddha offered a radical alternative to empire building - a way of living that had no place for violence or hierarchy and was open to everyone. Great thinkers from Socrates to Confucius proposed new ideas about how to rule more wisely and live in a better society. And in Greece, democracy was born - the greatest political experiment of all. But within just a few years, its future would be under threat from invasion by an empire in the east...

As the early June sun begins to set over a calm river in Central Hungary, masses of ghostly shapes emerge from their larval cases to take to the air for the first time. They are mayflies and in a spectacular display, thousands of them demonstrate how the very first wings were used. From the stunning aerobatics of hoverflies in an English garden to the mass migration of purple crow butterflies in the valleys of Taiwan, this episode tells the tale of the first animals ever to take to the air. Unique footage reveals the lightning fast reactions of bluebottles and hoverflies, filmed with one of the world's fastest cameras, and Sir David Attenborough handles the world's largest (and perhaps most ferocious) insect - the Titan beetle.

Marine invertebrates are some of the most bizarre and beautiful animals on the planet, and thrive in the toughest parts of the oceans. Divers swim into a shoal of predatory Humboldt squid as they emerge from the ocean depths to hunt in packs. When cuttlefish gather to mate, their bodies flash in stroboscopic colours. Time-lapse photography reveals thousands of starfish gathering under the Arctic ice to devour a seal carcass. A giant octopus commits suicide for her young. A camera follows her into a cave which she walls up, then she protects her eggs until she starves. The greatest living structures on earth, coral reefs, are created by tiny animals in some of the world's most inhospitable waters.