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.

During the chaotic final weeks of the Vietnam War, the North Vietnamese Army closes in on Saigon as the panicked South Vietnamese people desperately attempt to escape. On the ground, American soldiers and diplomats confront the same moral quandary: whether to obey White House orders to evacuate U.S. citizens only--or to risk treason and save the lives of as many South Vietnamese citizens as they can.

More than 2,000 years ago, the thriving city of Petra rose up in the bone-dry desert of what is now Jordan. An oasis of culture and abundance, the city was built by wealthy merchants whose camel caravans transported incense and spices from the Arabian Gulf. They carved spectacular temple-tombs into its soaring cliffs, raised a monumental Great Temple at its heart, and devised an ingenious system that channeled water to vineyards, bathhouses, fountains, and pools. But following a catastrophic earthquake and a slump in its desert trade routes, Petra's unique culture faded and was lost to most of the world for nearly a thousand years. Now, in a daring experiment, an archaeologist and sculptors team up to carve an iconic temple-tomb to find out how the ancient people of Petra built their city of stone. Meanwhile, scientists using remote sensors and hydraulic flumes uncover the vast city and its sophisticated water system. The race is on to discover how these nomads created this oasis of culture in one of the harshest climates on Earth.

David Attenborough asks three key questions: how and why did Darwin come up with his theory of evolution? Why do we think he was right? And why is it more important now than ever before? David starts his journey in Darwin's home at Down House in Kent, where Darwin worried and puzzled over the origins of life. David goes back to his roots in Leicestershire, where he hunted for fossils as a child, and where another schoolboy unearthed a significant find in the 1950s. And he revisits Cambridge University, where both he and Darwin studied, and where many years later the DNA double helix was discovered, providing the foundations for genetics. At the end of his journey in the Natural History Museum in London, David concludes that Darwin's great insight revolutionised the way in which we see the world. We now understand why there are so many different species, and why they are distributed in the way they are. But above all, Darwin has shown us that we are not set apart from the natural world, and do not have dominion over it. We are subject to its laws and processes, as are all other animals on earth to which, indeed, we are related.

Sagan discusses comets and asteroids as planetary impactors, giving recent examples of the Tunguska event and a lunar impact described by Canterbury monks in 1178. It moves to a description of the environment of Venus, from the previous fantastic theories of people such as Immanuel Velikovsky to the information gained by the Venera landers and its implications for Earth's greenhouse effect. The Cosmos Update highlights the connection to global warming.

The magnificent ancient city of pyramids at Caral in Peru is a thousand years older than the earliest known civilisation in the Americas and, at 2,627 BC, is as old as the pyramids of Egypt. Many now believe it is the fabled missing link of archaeology - a 'mother city'. If so, then these extraordinary findings could finally answer one of the great questions of archaeology: why did humans become civilised?" For over a century, archaeologists have been searching for what they call a mother city. Civilisation began in only six areas of the world: Egypt, Mesopotamia, India, China, Peru and Central America. In each of these regions people moved from small family units to build cities of thousands of people. They crossed the historic divide, one of the great moments in human history. Why? To find the answer archaeologists needed to find a mother city - the first stage of city-building. Caral, is so much older than anything else in South America that it is a clear candidate to be the mother city. It also is in pristine condition. Nothing has been built on it at all. Instead laid out before the world is an elaborate complex of pyramids, temples, an amphitheatre and ordinary houses. Scientists developed a number of theories. Some said it was because of the development of trade, others that it was irrigation. Some even today believe it was all because of aliens. Gradually an uneasy consensus emerged. The key force common to all civilisations was warfare. Crucially, there is not the faintest trace of warfare at Caral; no battlements, no weapons, no mutilated bodies. Instead, Ruth's findings suggest it was a gentle society, built on commerce and pleasure. In one of the pyramids they uncovered beautiful flutes made from condor and pelican bones. They have also found evidence of a culture that took drugs and perhaps aphrodisiacs. Most stunning of all, they have found the remains of a baby, lovingly wrapped and buried with a precious necklace made of stone beads.