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