Paul Erdős (1913 – 1996) is one of the most celebrated mathematicians of the 20th century. During his long career, he made a number of impressive advances in our understanding of maths and developed whole new fields in the subject. He was born into a Jewish family in Hungary just before the outbreak of World War I, and his life was shaped by the rise of fascism in Europe, anti-Semitism and the Cold War. His reputation for mathematical problem solving is unrivalled and he was extraordinarily prolific. He produced more than 1,500 papers and collaborated with around 500 other academics. He also had an unconventional lifestyle. Instead of having a long-term post at one university, he spent much of his life travelling around visiting other mathematicians, often staying for just a few days. With Colva Roney-Dougal Professor of Pure Mathematics at the University of St Andrews Timothy Gowers Professor of Mathematics at the College de France in Paris and Fellow of Trinity College, Cambridge and Andrew Treglown Associate Professor in Mathematics at the University of Birmingham The image above shows a graph occurring in Ramsey Theory. It was created by Dr Katherine Staden, lecturer in the School of Mathematics at the Open University.

Melvyn Bragg and guests discuss the pioneering Danish astronomer Tycho Brahe (1546 – 1601) whose charts offered an unprecedented level of accuracy. In 1572 Brahe's observations of a new star challenged the idea, inherited from Aristotle, that the heavens were unchanging. He went on to create his own observatory complex on the Danish island of Hven, and there, working before the invention of the telescope, he developed innovative instruments and gathered a team of assistants, taking a highly systematic approach to observation. A second, smaller source of renown was his metal prosthetic nose, which he needed after a serious injury sustained in a duel. The image above shows Brahe aged 40, from the Atlas Major by Johann Blaeu. With Ole Grell Emeritus Professor in Early Modern History at the Open University Adam Mosley Associate Professor of History at Swansea University and Emma Perkins Affiliate Scholar in the Department of History and Philosophy of Science at the University of Cambridge.

Melvyn Bragg and guests discuss the discovery made in 1911 by the Dutch physicist Heike Kamerlingh Onnes (1853-1926). He came to call it Superconductivity and it is a set of physical properties that nobody predicted and that none, since, have fully explained. When he lowered the temperature of mercury close to absolute zero and ran an electrical current through it, Kamerlingh Onnes found not that it had low resistance but that it had no resistance. Later, in addition, it was noticed that a superconductor expels its magnetic field. In the century or more that has followed, superconductors have already been used to make MRI scanners and to speed particles through the Large Hadron Collider and they may perhaps bring nuclear fusion a little closer (a step that could be world changing). The image above is from a photograph taken by Stephen Blundell of a piece of superconductor levitating above a magnet. With Nigel Hussey Professor of Experimental Condensed Matter Physics at the University of Bristol and Radbout University Suchitra Sebastian Professor of Physics at the Cavendish Laboratory at the University of Cambridge And Stephen Blundell Professor of Physics at the University of Oxford and Fellow of Mansfield College Producer: Simon Tillotson

Melvyn Bragg and guests discuss the voyage of HMS Challenger which set out from Portsmouth in 1872 with a mission a to explore the ocean depths around the world and search for new life. The scale of the enterprise was breath taking and, for its ambition, it has since been compared to the Apollo missions. The team onboard found thousands of new species, proved there was life on the deepest seabeds and plumbed the Mariana Trench five miles below the surface. Thanks to telegraphy and mailboats, its vast discoveries were shared around the world even while Challenger was at sea, and they are still being studied today, offering insights into the ever-changing oceans that cover so much of the globe and into the health of our planet. The image above is from the journal of Pelham Aldrich R.N. who served on the Challenger Surveying Expedition from 1872-5. With Erika Jones Curator of Navigation and Oceanography at Royal Museums Greenwich Sam Robinson Southampton Marine and Maritime Institute Research Fellow at the University of Southampton And Giles Miller Principal Curator of Micropalaeontology at the Natural History Museum London Producer: Simon Tillotson

Melvyn Bragg and guests discuss one of the greatest changes in the history of life on Earth. Around 400 million years ago some of our ancestors, the fish, started to become a little more like humans. At the swampy margins between land and water, some fish were turning their fins into limbs, their swim bladders into lungs and developed necks and eventually they became tetrapods, the group to which we and all animals with backbones and limbs belong. After millions of years of this transition, these tetrapod descendants of fish were now ready to leave the water for a new life of walking on land, and with that came an explosion in the diversity of life on Earth. The image above is a representation of Tiktaalik Roseae, a fish with some features of a tetrapod but not one yet, based on a fossil collected in the Canadian Arctic. With Emily Rayfield Professor of Palaeobiology at the University of Bristol Michael Coates Chair and Professor of Organismal Biology and Anatomy at the University of Chicago And Steve Brusatte Professor of Palaeontology and Evolution at the University of Edinburgh Producer: Simon Tillotson