Melvyn Bragg and guests discuss the history of optics. From telescopes to microscopes, from star-gazing to the intimacies of a magnified flea. As Galileo turned his telescope to the heavens in the early 1600s, Kepler began to formulate a theory of optics. The new and improving instruments went hand in hand with radical new ideas about how we see and what we see. Spectacles allowed scholars to study long into the evening (and into old age), while giant telescopes, up to 100 feet long, led to the discovery of planets and attempts to map the universe. The craze for optical trickery swept Europe with enthusiastic amateurs often providing valuable discoveries. But this new view of the world through a lens raised questions too – how much can you rely on the senses, on what you see? The further into space you can spy, the larger and more unmanageable the universe becomes. At the same time, the microscope was utterly transforming the world close at hand.So how did these developments inform ideas of knowledge? If new methods of scientific observation support an empirical approach, what does this mean for divine, innate reason?With Simon Schaffer, Professor in History and Philosophy of Science at the University of Cambridge; Jim Bennett, Director of the Museum of the History of Science and Fellow of Linacre College at the University of Oxford; Emily Winterburn, Curator of Astronomy at the National Maritime Museum

Melvyn Bragg and guests discuss one of the most important philosophers of the 20th century, Karl Popper whose ideas about science and politics robustly challenged the accepted ideas of the day. He strongly resisted the prevailing empiricist consensus that scientists' theories could be proved true.Popper wrote: “The more we learn about the world and the deeper our learning, the more conscious, specific and articulate will be our knowledge of what we do not know, our knowledge of our ignorance”. He believed that even when a scientific principle had been successfully and repeatedly tested, it was not necessarily true. Instead it had simply not proved false, yet! This became known as the theory of falsification.He called for a clear demarcation between good science, in which theories are constantly challenged, and what he called “pseudo sciences” which couldn't be tested. His debunking of such ideologies led some to describe him as the “murderer of Freud and Marx”. He went on to apply his ideas to politics, advocating an Open Society. His ideas influenced a wide range of politicians, from those close to Margaret Thatcher, to thinkers in the Eastern Communist bloc and South America.So how did Karl Popper change our approach to the philosophy of science? How have scientists and philosophers made use of his ideas? And how are his theories viewed today? Are we any closer to proving scientific principles are “true”?With John Worrall, Professor of Philosophy of Science at the London School of Economics; Anthony O'Hear, Weston Professor of Philosophy at Buckingham University; Nancy Cartwright, Professor of Philosophy at the LSE and the University of California

Melvyn Bragg and guests discuss the Greek mathematician Archimedes. Reputed to have shouted “Eureka!” as he leapt from his bath having discovered the principles of floating bodies. Whatever the truth of the myths surrounding the man, he was certainly one of the world’s great mathematicians. The practical application of his work in pulleys and levers created formidable weapons such as catapults and ship tilting systems, allowing his home city in Sicily to defend itself against the Romans. “Give me a place to stand and I will move the earth”, he declared.But despite these triumphs, his true love remained maths for maths sake. Plutarch writes: “He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life.” His most important breakthroughs came in the field of geometry with his work on the areas and volumes of curved objects.So how did this Greek mathematician in the third century BC arrive at a calculation of Pi? Did he really create a Death Ray to fight off invading ships? And what does a recently discovered manuscript reveal about his methods?With Jackie Stedall, Junior Research Fellow in the History of Mathematics at Queen's College, Oxford; Serafina Cuomo, Reader in the History of Science at Imperial College London; George Phillips, Honorary Reader in Mathematics at St Andrews University

Melvyn Bragg and guests discuss the Jesuits, a Catholic religious order of priests who became known as “the school masters of Europe”. Founded in the 16th century by the soldier Ignatius Loyola, they became a major force throughout the world, from China to South America. “Give us a boy and we will return you a man, a citizen of his country and a child of God”, they declared. By the 17th century there were more than 500 schools established across Europe. Their ideas about a standardised curriculum and teaching became the basis for many education systems today.They were also among the greatest patrons of art in early modern Europe, using murals and theatre to get their message across. To their enemies they were a sinister collective whose influence reached into the courts of kings. Their wealth and their adaptability to local customs abroad provoked suspicion, prompting their eventual suppression in the late 18th century. They were re-established in 1814 and now have more than twenty thousand members.So why was education so important to the Jesuit movement? How much influence did they really have in the courts and colonies of Europe? And were they really at the heart of conspiracies to murder kings?With Nigel Aston, Reader in Early Modern History at the University of Leicester; Simon Ditchfield, Reader in History at the University of York; Dame Olwen Hufton, Emeritus Fellow of Merton College, Oxford.

Melvyn Bragg and guests discuss the planet Mars. Named after the Roman god of war, Mars has been a source of continual fascination. It is one of our nearest neighbours in space, though it takes about a year to get there. It is very inhospitable with high winds racing across extremely cold deserts. But it is spectacular, with the highest volcano in the solar system and a giant chasm that dwarfs the Grand Canyon.For centuries there has been fierce debate about whether there is life on Mars and from the 19th century it was even thought there might be a system of canals on the planet. This insatiable curiosity has been fuelled by writers like HG Wells and CS Lewis and countless sci-fi films about little green men.So what do we know about Mars – its conditions, now and in the past? What is the evidence that there might be water and thus life on Mars? And when might we expect man to walk on its surface?With John Zarnecki, Professor of Space Science at the Open University and a team leader on the ExoMars mission; Colin Pillinger, Professor of Planetary Sciences at the Open University and leader of the Beagle 2 expedition to Mars; Monica Grady, Professor of Planetary and Space Sciences at the Open University and an expert on Martian meteorites.

Melvyn Bragg and guests discuss the contribution Indian mathematicians have made to our understanding of the subject. Mathematics from the Indian subcontinent has provided foundations for much of our modern thinking on the subject. They were thought to be the first to use zero as a number. Our modern numerals have their roots there too. And mathematicians in the area that is now India, Pakistan and Bangladesh were grappling with concepts such as infinity centuries before Europe got to grips with it. There’s even a suggestion that Indian mathematicians discovered Pythagoras’ theorem before Pythagoras. Some of these advances have their basis in early religious texts which describe the geometry necessary for building falcon-shaped altars of precise dimensions. Astronomical calculations used to decide the dates of religious festivals also encouraged these mathematical developments. So how were these advances passed on to the rest of the world? And why was the contribution of mathematicians from this area ignored by Europe for centuries?With George Gheverghese Joseph, Honorary Reader in Mathematics Education at Manchester University; Colva Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews; Dennis Almeida, Lecturer in Mathematics Education at Exeter University and the Open University.

Melvyn Bragg and guests discuss the speed of light. Scientists and thinkers have been fascinated with the speed of light for millennia. Aristotle wrongly contended that the speed of light was infinite, but it was the 17th Century before serious attempts were made to measure its actual velocity – we now know that it’s 186,000 miles per second. Then in 1905 Einstein’s Special Theory of Relativity predicted that nothing can travel faster than the speed of light. This then has dramatic effects on the nature of space and time. It’s been thought the speed of light is a constant in Nature, a kind of cosmic speed limit, now the scientists aren’t so sure. With John Barrow, Professor of Mathematical Sciences and Gresham Professor of Astronomy at Cambridge University; Iwan Morus, Senior Lecturer in the History of Science at The University of Wales, Aberystwyth; Jocelyn Bell Burnell, Visiting Professor of Astrophysics at Oxford University.

Melvyn Bragg and guests discuss the Poincaré Conjecture. The great French mathematician Henri Poincaré declared: “The scientist does not study mathematics because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing and life would not be worth living. And it is because simplicity, because grandeur, is beautiful that we preferably seek simple facts, sublime facts, and that we delight now to follow the majestic course of the stars.” Poincaré’s ground-breaking work in the 19th and early 20th century has indeed led us to the stars and the consideration of the shape of the universe itself. He is known as the father of topology – the study of the properties of shapes and how they can be deformed. His famous Conjecture in this field has been causing mathematicians sleepless nights ever since. He is also credited as the Father of Chaos Theory.So how did this great polymath change the way we understand the world and indeed the universe? Why did his conjecture remain unproved for almost a century? And has it finally been cracked?With June Barrow-Green, Lecturer in the History of Mathematics at the Open University; Ian Stewart, Professor of Mathematics at the University of Warwick; Marcus du Sautoy, Professor of Mathematics at the University of Oxford.

Melvyn Bragg and guests discuss the Needham Question; why Europe and not China developed modern technology. What do these things have in common? Fireworks, wood-block printing, canal lock-gates, kites, the wheelbarrow, chain suspension bridges and the magnetic compass. The answer is that they were all invented in China, a country that, right through the Middle Ages, maintained a cultural and technological sophistication that made foreign dignitaries flock to its imperial courts for trade and favour. But then, around 1700, the flow of ingenuity began to dry up and even reverse as Europe bore the fruits of the scientific revolution back across the globe. Why did Modern Science develop in Europe when China seemed so much better placed to achieve it? This is called the Needham Question, after Joseph Needham, the 20th century British Sinologist who did more, perhaps, than anyone else to try and explain it.But did Joseph Needham give a satisfactory answer to the question that bears his name? Why did China’s early technological brilliance not lead to the development of modern science and how did momentous inventions like gunpowder and printing enter Chinese society with barely a ripple and yet revolutionise the warring states of Europe? With Chris Cullen, Director of the Needham Research Institute in Cambridge; Tim Barrett, Professor of East Asian History at SOAS; Frances Wood, Head of Chinese Collections at the British Library.

Melvyn Bragg and guests discuss the Prussian naturalist and explorer Alexander Von Humboldt. He was possibly the greatest and certainly one of the most famous scientists of the 19th century. Darwin described him as 'the greatest scientific traveller who ever lived'. Goethe declared that one learned more from an hour in his company than eight days of studying books and even Napoleon was reputed to be envious of his celebrity.A friend of Goethe and an influence on Coleridge and Shelly, when Darwin went voyaging on the Beagle it was Humboldt's works he took for inspiration and guidance. At the time of his death in 1859, the year Darwin published On the Origin of Species, Humboldt was probably the most famous scientist in Europe. Add to this shipwrecks, homosexuality and Spanish American revolutionary politics and you have the ingredients for one of the more extraordinary lives lived in Europe (and elsewhere) in the 18th and 19th centuries. But what is Humboldt's true position in the history of science? How did he lose the fame and celebrity he once enjoyed and why is he now, perhaps, more important than he has ever been? With Jason Wilson, Professor of Latin American Literature at University College London, Patricia Fara, Affiliated Lecturer in the Department of History and Philosophy of Science at the University of Cambridge, Jim Secord, Professor in the Department of History and Philosophy of Science at the University of Cambridge and Director of the Darwin Correspondence Project.