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.

Professor Brian Cox journeys across the vastness of time and space revealing epic moments of sheer drama that changed the universe forever.
The series begins this epic exploration of the cosmos with a hymn to the great luminous bodies that bring light and warmth to the universe: the stars. It is estimated that there are two hundred trillion stars in the universe, each playing their part in an epic story of creation- a great saga that stretches from the dawn of time, with the arrival of the first star, through diverse generations until the arrival of our own star, the sun, and a civilization that has grown up in its light.

In 2014, the International Space Station had to move three times to avoid lethal chunks of space debris and there is an increasing problem of satellites mysteriously breaking down. With first-hand accounts from astronauts and experts, know the scale of the problem of space junk. Our planet is surrounded by hundreds of millions of pieces of junk moving at 17,000 miles per hour. Now the US government is investing a billion dollars to track them, and companies around the world are developing ways to clear up their mess - from robot arms to nets and harpoons. Learn the science behind the hit film Gravity and discovers the reality is far more worrying than the Hollywood fiction.

Perhaps the most mysterious structure on Earth, Stonehenge has stood on a plain in Southern England for 5000 years. Its foundations predate the Great Pyramids. It is one of mankind's most ancient mysteries. Why is it here? Is it a temple? A burial ground? A place for sacrifice? Or could the mystery of Stonehenge be revealed in its builders' desire to explore the unknown heavens and touch the universe? Using the cutting-edge computer-generated imagery that takes us into deep space, we'll also go inside a virtual Stonehenge to see what the ancients saw and push this prehistoric marvel to give up its age-old secrets.
In this episode we explore the possibility that this was a prehistoric astronomical observatory. Here ancient astronomer priests may have divined the complex movements of the Sun and Moon, recognizing patterns that would not be discovered elsewhere for thousands of years. The primitive Shamans may have also been the first astronomers to predict eclipses.

Dan Snow uncovers the lost Vikings in America with space archaeologist Dr Sarah Parcak. Sarah uses satellites 383 miles above the earth to spot ruins as small as 30cm buried beneath the surface. As Sarah searches for Viking sites from Britain to America, Dan explores how they voyaged thousands of miles when most ships never left the shoreline. He also tracks their expansion west, first as raiders and then as settlers and traders throughout Britain and beyond to Iceland and Greenland. In North America they excavate what could be the most westerly Viking settlement ever discovered.

This episode explores the nature of the greenhouse effect (discovered by Joseph Fourier and Svante Arrhenius), and the evidence demonstrating the existence of global warming from humanity's influence. Tyson begins by describing the long-term history of the planet Venus; based on readings from the Venera series of probes to the planet, the planet had once had an ocean and an atmosphere, but due to the release of carbon dioxide from volcanic eruptions, the runaway greenhouse effect on Venus caused the surface temperatures to increase and boiled away the oceans. Tyson then notes the delicate nature of the amount of carbon dioxide in the atmosphere can influence Earth's climate due to the greenhouse effect, and that levels of carbon dioxide have been increasing since the start of the 20th century. Evidence has shown this to be from mankind's consumption of oil, coal, and gas instead of from volcanic eruptions due to the isotopic signature of the carbon dioxide. The increase in carbon dioxide has led to an increase in temperatures, in turn leading to positive feedback loops of the melting polar ice caps and dethawing of the permafrost to increase carbon dioxide levels. Tyson then notes that humans have discovered means of harvesting solar power, such as Augustin Mouchot's solar-driven motor in the 19th century, and Frank Shuman's solar-based steam generator in the 1910's. Tyson points out that in both cases, the economics and ease of using cheap coal and oil caused these inventions to be overlooked at the time. Today, solar and wind-power systems would be able to collect enough solar energy from the sun easily. Tyson then compares the motivation for switching to these cleaner forms of energy to the efforts of the Space race and emphasizes that it is not too late for humanity to correct its course.