Melvyn Bragg and guests discuss The Battle of Lepanto, 1571, the last great sea battle between galleys, in which the Catholic fleet of the Holy League of principally Venice, Spain, the Papal States, Malta, Genoa, and Savoy defeated the Ottoman forces of Selim II. When much of Europe was divided over the Reformation, this was the first major victory of a Christian force over a Turkish fleet. The battle followed the Ottoman invasion of Venetian Cyprus and decades in which the Venetians had been trying to stop the broader westward expansion of the Ottomans into the Mediterranean. The outcome had a great impact on morale in Europe and Pope Pius V established a feast day of Our Lady of Victory. Some historians call it the most significant sea battle since Actium (31 BC). However, the Ottomans viewed the loss as less significant than their victory in Cyprus and, within two years, the Holy League had broken up.WithDiarmaid MacCulloch Professor of the History of the Church at the University of OxfordKate Fleet Director of the Skilliter Centre for Ottoman Studies and Fellow of Newnham College, University of CambridgeAndNoel Malcolm A Senior Research Fellow in History at All Soul's College, University of Oxford Producer: Simon Tillotson.

Melvyn Bragg and guests discuss the problem of P versus NP, which has a bearing on online security. There is a $1,000,000 prize on offer from the Clay Mathematical Institute for the first person to come up with a complete solution. At its heart is the question "are there problems for which the answers can be checked by computers, but not found in a reasonable time?" If the answer to that is yes, then P does not equal NP. However, if all answers can be found easily as well as checked, if only we knew how, then P equals NP. The area has intrigued mathematicians and computer scientists since Alan Turing, in 1936, found that it's impossible to decide in general whether an algorithm will run forever on some problems. Resting on P versus NP is the security of all online transactions which are currently encrypted: if it transpires that P=NP, if answers could be found as easily as checked, computers could crack passwords in moments.With Colva Roney-Dougal Reader in Pure Mathematics at the University of St AndrewsTimothy Gowers Royal Society Research Professor in Mathematics at the University of CambridgeAnd Leslie Ann Goldberg Professor of Computer Science and Fellow of St Edmund Hall, University of OxfordProducer: Simon Tillotson.

Melvyn Bragg and guests discuss the Empire of Mali which flourished from 1200 to 1600 and was famous in the wider world for the wealth of rulers such as Mansa Musa. Mali was the largest empire in west Africa and for almost 400 years controlled the flow of gold from mines in the south up to the Mediterranean coast and across to the Middle East. These gold mines were the richest known deposits in the 14th Century and produced around half of the world's gold. When Mansa Musa journeyed to Cairo in 1324 as part of his Hajj, he distributed so much gold that its value depreciated by over 10%. Some of the mosques he built on his return survive, albeit rebuilt, such as the UNESCO World Heritage Site of the Great Mosque of Djenne. With Amira Bennison Reader in the History and Culture of the Maghrib at the University of CambridgeMarie Rodet Senior Lecturer in the History of Africa at SOASAndKevin MacDonald Professor of African Archaeology Chair of the African Studies Programme at University College, London Producer: Simon Tillotson.

Melvyn Bragg and guests discuss Simone de Beauvoir. "One is not born, but rather becomes, a woman," she wrote in her best known and most influential work, The Second Sex, her exploration of what it means to be a woman in a world defined by men. Published in 1949, it was an immediate success with the thousands of women who bought it. Many male critics felt men came out of it rather badly. Beauvoir was born in 1908 to a high bourgeois family and it was perhaps her good fortune that her father lost his money when she was a girl. With no dowry, she pursued her education in Paris to get work and in a key exam to allow her to teach philosophy, came second only to Jean Paul Sartre. He was retaking. They became lovers and, for the rest of their lives together, intellectual sparring partners. Sartre concentrated on existentialist philosophy; Beauvoir explored that, and existentialist ethics, plus the novel and, increasingly in the decades up to her death in 1986, the situation of women in the world. With Christina Howells Professor of French and Fellow of Wadham College at the University of OxfordMargaret Atack Professor of French at the University of LeedsAnd Ursula Tidd Professor of Modern French Literature and Thought at the University of ManchesterProducer: Simon Tillotson.

Melvyn Bragg and guests discuss the life and work of Hans Holbein the Younger (1497-1543) during his two extended stays in England, when he worked at the Tudor Court and became the King's painter. Holbein created some of the most significant portraits of his age, including an image of Henry VIII, looking straight at the viewer, hands on hips, that has dominated perceptions of him since. The original at Whitehall Palace was said to make visitors tremble at its majesty. Holbein was later sent to Europe to paint the women who might be Henry's fourth wife; his depiction of Anne of Cleves was enough to encourage Henry to marry her, a decision Henry quickly regretted and for which Thomas Cromwell, her supporter, was executed. His paintings still shape the way we see those in and around the Tudor Court, including Cromwell, Thomas More, the infant Prince Edward (of which there is a detail, above), The Ambassadors and, of course, Henry the Eighth himself.WithSusan Foister Curator of Early Netherlandish, German and British Painting at the National GalleryJohn Guy A fellow of Clare College, University of CambridgeAndMaria Hayward Professor of Early Modern History at the University of SouthamptonProducer: Simon Tillotson.

Alexander the Great is one of the most celebrated military commanders in history. Born into the Macedonian royal family in 356 BC, he gained control of Greece and went on to conquer the Persian Empire, defeating its powerful king, Darius III. At its peak, Alexander's empire covered modern Turkey, Syria, Egypt, Iran, Iraq, Afghanistan, Pakistan and part of India. As a result, Greek culture and language was spread into regions it had not penetrated before, and he is also remembered for founding a number of cities. Over the last 2,000 years, the legend of Alexander has grown and he has influenced numerous generals and politicians.With:Paul Cartledge Emeritus Professor of Greek Culture and AG Leventis Senior Research Fellow at Clare College, University of CambridgeDiana Spencer Professor of Classics at the University of BirminghamRachel Mairs Lecturer in Classics at the University of ReadingProducer: Victoria Brignell.

Melvyn Bragg and guests discuss the rise of the idea of perpetual motion and its decline, in the 19th Century, with the Laws of Thermodynamics. For hundreds of years, some of the greatest names in science thought there might be machines that could power themselves endlessly. Leonardo Da Vinci tested the idea of a constantly-spinning wheel and Robert Boyle tried to recirculate water from a draining flask. Gottfried Leibniz supported a friend, Orffyreus, who claimed he had built an ever-rotating wheel. An increasing number of scientists voiced their doubts about perpetual motion, from the time of Galileo, but none could prove it was impossible. For scientists, the designs were a way of exploring the laws of nature. Others claimed their inventions actually worked, and promised a limitless supply of energy. It was not until the 19th Century that the picture became clearer, with the experiments of James Joule and Robert Mayer on the links between heat and work, and the establishment of the First and Second Laws of Thermodynamics.With Ruth Gregory Professor of Mathematics and Physics at Durham UniversityFrank Close Professor Emeritus of Physics at the University of OxfordandSteven Bramwell Professor of Physics and former Professor of Chemistry at University College LondonProducer: Simon Tillotson.

Born near Mexico City in 1907, Frida Kahlo is considered one of Mexico's greatest artists. She took up painting after a bus accident left her severely injured, was a Communist, married Diego Rivera, a celebrated muralist, became friends with Trotsky and developed an iconic series of self-portraits. Her work brings together elements such as surrealism, pop culture, Aztec and Indian mythology and commentary on Mexican culture. In 1938, artist and poet Andre Breton organised an exhibition of her work in New York, writing in the catalogue, "The Art of Frida Kahlo is a ribbon around a bomb." She was not as widely appreciated during her lifetime as she has since become, but is now one of the most recognised artists of the 20th century. WithPatience Schell Chair in Hispanic Studies at the University of AberdeenValerie Fraser Emeritus Professor of Latin American Art at the University of EssexAndAlan Knight Emeritus Professor of the History of Latin America at the University of OxfordProducer: Simon Tillotson.

Frederick the Great ruled Prussia from 1740 until his death in 1786. Born in 1712, he increased the power of the state, he made Prussia the leading military power in Europe and his bold campaigns had great implications for the European political landscape. An absolute monarch in the age of enlightenment, he was a prolific writer, attracted figures such as Voltaire to his court, fostered education and put Berlin firmly on the cultural map. He was much admired by Napoleon and was often romanticised by German historians, becoming a hero for many in united Germany in the 19th and 20th centuries. Others, however, vilified him for aspects such as his militarism and the partition of Poland. With Tim Blanning Emeritus Professor of Modern European History at the University of CambridgeKatrin Kohl Professor of German Literature at the University of Oxford and a Fellow of Jesus CollegeAnd Thomas Biskup Lecturer in Early Modern History at the University of HullProducer: Simon Tillotson.

In 1977, scientists in the submersible "Alvin" were exploring the deep ocean bed off the Galapagos Islands. In the dark, they discovered hydrothermal vents, like chimneys, from which superheated water flowed. Around the vents there was an extraordinary variety of life, feeding on microbes which were thriving in the acidity and extreme temperature of the vents. While it was already known that some microbes are extremophiles, thriving in extreme conditions, such as the springs and geysers of Yellowstone Park (pictured), that had not prepared scientists for what they now found. Since the "Alvin" discovery, the increased study of extremophile microbes has revealed much about what is and is not needed to sustain life on Earth and given rise to new theories about how and where life began. It has also suggested forms and places in which life might be found elsewhere in the Universe. With Monica Grady Professor of Planetary and Space Sciences at the Open UniversityIan Crawford Professor of Planetary Science and Astrobiology at Birkbeck University of LondonAndNick Lane Reader in Evolutionary Biochemistry at University College LondonProducer: Simon Tillotson.