Melvyn Bragg and guests discuss Renaissance Mathematics. As with so many areas of European thought, mathematics in the Renaissance was a question of recovering and, if you were very lucky, improving upon Greek ideas. The geometry of Euclid, Appollonius and Ptolemy ruled the day. Yet within two hundred years, European mathematics went from being an art that would unmask the eternal shapes of geometry to a science that could track the manifold movements and changes of the real world. The Arabic tradition of Algebra was also assimilated. In its course it changed the way people understood numbers, movement, time, even nature itself and culminated in the calculus of Isaac Newton and Gottfried Leibniz. But how did this profound change come about? What were the ideas that drove it and is this the period in which mathematics became truly modern?With Robert Kaplan, co-founder of the Maths Circle at Harvard University; Jim Bennett, Director of the Museum of Science and Fellow of Linacre College, University of Oxford; Jackie Stedall, Research Fellow in the History of Mathematics, The Queen's College, Oxford.

Exploring the interconnectedness of the senses and questioning our perception of the world based on expectations. Evolution of perception and how the brain adds information to the senses. The dominance of sight over hearing and the importance of lip reading. Chemical senses of smell and taste working together. Experience of the physical world and the question of illusions. Overlooked senses of proprioception and vestibular system.

Melvyn Bragg and guests discuss 'dark energy'. Only 5% of our universe is composed of visible matter, stars, planets and people; something called 'dark matter' makes up about 25% and an enormous 70% of the universe is pervaded with the mysteriously named 'dark energy'. It is a recent discovery and may be only a conjecture, but it has been invoked to explain an abiding riddle of the cosmos: if the expansion of the universe is powered by the energy of the Big Bang, then why isn't the expansion slowing down over time as the initial energy runs down and the attractive force of gravity asserts itself? Scientists had predicted a Big Crunch as the logical opposite of the Big Bang, but far from retracting, the expansion of the universe is actually accelerating...it's running away with itself.How do we know that the universe is behaving like this and what's causing it? If dark energy is the culprit, then what is this elusive, though omnipresent entity?With Sir Martin Rees, Astronomer Royal and Professor of Cosmology and Astrophysics, Cambridge University; Carolin Crawford, Royal Society University Research Fellow at the Institute of Astronomy, University of Cambridge; Sir Roger Penrose, Emeritus Rouse Ball Professor of Maths at Oxford University.

Melvyn Bragg and guests discuss the history of Alchemy, the ancient science of transformations. The most famous alchemical text is the Emerald Tablet, written around 500BC and attributed to the mythical Egyptian figure of Hermes Trismegistus. Among its twelve lines are the essential words - “as above, so below". They capture the essence of alchemy, that the heavens mirror the earth and that all things correspond to one another. Alchemy was taken up by some of the most extraordinary people in our intellectual development, including Roger Bacon, Paracelsus, the father of chemistry, Robert Boyle, and, most famously, Isaac Newton, who wrote more about alchemy than he did about physics. It is now contended that it was Newton’s studies into alchemy which gave him the fundamental insight into the famous three laws of motion and gravity.With Peter Forshaw, Lecturer in Renaissance Philosophies at Birkbeck, University of London, Lauren Kassell, Lecturer in the History and Philosophy of Science at the University of Cambridge, Stephen Pumfrey, Senior Lecturer in the History of Science at the University of Lancaster.

Melvyn Bragg and guests discuss the Cambrian period when there was an explosion of life on Earth. In the Selkirk Mountains of British Columbia in Canada, there is an outcrop of limestone shot through with a seam of fine dark shale. A sudden mudslide into shallow water some 550 million years ago means that a startling array of wonderful organisms has been preserved within it. Wide eyed creatures with tentacles below and spines on their backs, things like flattened rolls of carpet with a set of teeth at one end, squids with big lobster-like arms. There are thousands of them and they seem to testify to a time when evolution took a leap and life on this planet suddenly went from being small, simple and fairly rare to being large, complex, numerous and dizzyingly diverse. It happened in the Cambrian Period and it's known as the Cambrian Explosion.But if this is the great crucible of life on Earth, what could have caused it? How do the strange creatures relate to life as we see it now? And what does the Cambrian Explosion tell us about the nature of evolution?With Simon Conway Morris, Professor of Evolutionary Palaeobiology, Cambridge University; Richard Corfield, Visiting Senior Lecturer at the Centre for Earth, Planetary, Space and Astronomical Research, Open University; Jane Francis, Professor of Palaeoclimatology, University of Leeds.

Melvyn Bragg and guests discuss the mind/body problem in philosophy. At the start of René Descartes' Sixth Meditation he writes: "there is a great difference between mind and body, inasmuch as body is by nature always divisible, and mind is entirely indivisible. For when I consider the mind, or myself in so far as I am merely a thinking thing, I am unable to distinguish many parts within myself; I understand myself to be something quite single and complete. Although the whole mind seems to be united to the whole body, I recognize that if a foot or an arm or any other part of the body is cut off nothing has thereby been taken away from the mind".This thinking is the basis of what's known as 'Cartesian dualism', Descartes' attempt to address one of the central questions in philosophy, the mind/body problem: is the mind part of the body, or the body part of the mind? If they are distinct, then how do they interact? And which of the two is in charge?With Anthony Grayling, Reader in Philosophy at Birkbeck, University of London; Julian Baggini, editor of The Philosophers' Magazine; Sue James, Professor of Philosophy at Birkbeck, University of London.

Melvyn Bragg and guests discuss the Second Law of Thermodynamics which can be very simply stated like this: "Energy spontaneously tends to flow from being concentrated in one place to becoming diffused and spread out". It was first formulated – derived from ideas first put forward by Lord Kelvin - to explain how a steam engine worked, it can explain why a cup of tea goes cold if you don't drink it and how a pan of water can be heated to boil an egg.But its application has been found to be rather grander than this. The Second Law is now used to explain the big bang, the expansion of the cosmos and even suggests our inexorable passage through time towards the 'heat death' of the universe. It's been called the most fundamental law in all of science, and CP Snow in his Two Cultures wrote: "Not knowing the Second Law of Thermodynamics is like never having read a work of Shakespeare".What is the Second Law? What are its implications for time and energy in the universe, and does it tend to be refuted by the existence of life and the theory of evolution?With John Gribbin, Visiting Fellow in Astronomy at the University of Sussex; Peter Atkins, Professor of Chemistry at Oxford University; Monica Grady, Head of Petrology and Meteoritics at the Natural History Museum.

Melvyn Bragg and guests discuss the extraordinary mind of the psychiatrist Carl Gustav Jung. In 1907 Sigmund Freud met a young man and fell into a conversation that is reputed to have lasted for 13 hours. That man was the Swiss psychiatrist Carl Gustav Jung. Freud is celebrated as the great pioneer of the 20th century mind, but the idea that personality types can be 'introverted' or 'extroverted', that certain archetypal images and stories repeat themselves constantly across the collective history of mankind, and that personal individuation is the goal of life, all belong to Jung: "Your vision will become clear only when you look into your heart... Who looks outside, dreams. Who looks inside, awakens", he declared. And he also said "Show me a sane man and I will cure him for you".Who was Jung? What is the essence and influence of his thought? And how did he become such a controversial and, for many, such a beguiling figure?With Brett Kahr, Senior Clinical Research Fellow in Psychotherapy and Mental Health at the Centre for Child Mental Health in London and a practising Freudian; Ronald Hayman, writer and biographer of Jung; Andrew Samuels, Professor of Analytical Psychology at the University of Essex and a Jungian analyst in clinical practice.

Melvyn Bragg and guests discuss the Higgs Boson particle. One weekend in 1964 the Scottish scientist Peter Higgs was walking in the Cairngorm Mountains. On his return to his laboratory in Edinburgh the following Monday, he declared to his colleagues that he had just experienced his 'one big idea' and now had an answer to the mystery of how matter in the universe got its mass. That big idea took many years of refining, but it has now generated so much international interest and has such an important place in physics that well over one billion pounds is being spent in the hope that he was right. It's the biggest science project on Earth; the quest to find the 'Higgs Boson', a fundamental constituent of nature that - if it does exist - has such a central role in defining the universe that it's also known as the God Particle.What is the Higgs Boson? Why is it so important to scientists and how are they planning to find it?With Jim Al-Khalili, Senior Lecturer in Physics at the University of Surrey; David Wark, Professor of Experimental Physics at Imperial College London and the Rutherford Appleton Laboratory; Professor Roger Cashmore, former Research Director at CERN and now Principal of Brasenose College, Oxford.

Melvyn Bragg and guests discuss the dawn of the age of electricity. In Gulliver's Travels, published in 1726, Jonathan Swift satirised natural philosophers as trying to extract sunbeams from cucumbers. Perhaps he would have been surprised, or even horrified, by the sheer force of what these seemingly obscure experimentalists were about to unleash on society. Electricity soon reached into all areas of 18th century life, as Royal Society Fellows vied with showmen and charlatans to reveal its wonders to the world. It was, claimed one commentator, 'an entertainment for Angels rather than for Men'. Electricity also posed deep questions about the nature of life. For some it was the divine spark that animated all things, for others it represented a dangerous materialism that reduced humans to mere machines.But how did electricity develop in the 18th and 19th centuries? Why was it so politically contentious and how was it understood during the age in which it changed the world forever?With Simon Schaffer, Professor in History and Philosophy of Science at the University of Cambridge and a Fellow of Darwin College; Patricia Fara, historian of science and a Fellow of Clare College, Cambridge; Iwan Morus, Lecturer in the History of Science at Queen's University Belfast.