Melvyn Bragg and guests discuss Euclid's Elements, a mathematical text book attributed to Euclid and in use from its appearance in Alexandria, Egypt around 300 BC until modern times, dealing with geometry and number theory. It has been described as the most influential text book ever written. Einstein had a copy as a child, which he treasured, later saying "If Euclid failed to kindle your youthful enthusiasm, then you were not born to be a scientific thinker."

With

Marcus du Sautoy

Serafina Cuomo

And

June Barrow-Green

Producer: Simon Tillotson.

Melvyn Bragg and guests discuss the impact of the eruption of Mt Tambora, in 1815, on the Indonesian island of Sambawa. This was the largest volcanic eruption in recorded history and it had the highest death toll, devastating people living in the immediate area. Tambora has been linked with drastic weather changes in North America and Europe the following year, with frosts in June and heavy rains throughout the summer in many areas. This led to food shortages, which may have prompted westward migration in America and, in a Europe barely recovered from the Napoleonic Wars, led to widespread famine.

With

Clive Oppenheimer

Jane Stabler

And

Lawrence Goldman

Producer: Simon Tillotson.

Melvyn Bragg and guests discuss the neutron, one of the particles found in an atom's nucleus. Building on the work of Ernest Rutherford, the British physicist James Chadwick won the Nobel Prize for Physics for his discovery of the neutron in 1932. Neutrons play a fundamental role in the universe and their discovery was at the heart of developments in nuclear physics in the first half of the 20th century.

With

Val Gibson

Andrew Harrison

And

Frank Close

Melvyn Bragg and guests discuss the life and work of Robert Hooke (1635-1703) who worked for Robert Boyle and was curator of experiments at the Royal Society. The engraving of a flea, above, is taken from his Micrographia which caused a sensation when published in 1665. Sometimes remembered for his disputes with Newton, he studied the planets with telescopes and snowflakes with microscopes. He was an early proposer of a theory of evolution, discovered light diffraction with a wave theory to explain it and felt he was rarely given due credit for his discoveries.

With

David Wootton

Patricia Fara

And

Rob Iliffe

Producer: Simon Tillotson.

Melvyn Bragg and guests discuss the origins, development and uses of chromatography. In its basic form, it is familiar to generations of schoolchildren who put a spot of ink at the bottom of a strip of paper, dip it in water and then watch the pigments spread upwards, revealing their separate colours. Chemists in the 19th Century started to find new ways to separate mixtures and their work was taken further by Mikhail Tsvet, a Russian-Italian scientist who is often credited with inventing chromatography in 1900. The technique has become so widely used, it is now an integral part of testing the quality of air and water, the levels of drugs in athletes, in forensics and in the preparation of pharmaceuticals.

With

Andrea Sella

Apryll Stalcup

And

Leon Barron

Melvyn Bragg and guests discuss the planet Saturn with its rings of ice and rock and over 60 moons. In 1610, Galileo used an early telescope to observe Saturn, one of the brightest points in the night sky, but could not make sense of what he saw: perhaps two large moons on either side. When he looked a few years later, those supposed moons had disappeared. It was another forty years before Dutch scientist Christiaan Huygens solved the mystery, realizing the moons were really a system of rings. Successive astronomers added more detail, with the greatest leaps forward in the last forty years. The Pioneer 11 spacecraft and two Voyager missions have flown by, sending back the first close-up images, and Cassini is still there, in orbit, confirming Saturn, with its rings and many moons, as one of the most intriguing and beautiful planets in our Solar System.

With

Carolin Crawford

Michele Dougherty

And

Andrew Coates

Melvyn Bragg and guests discuss the eminent 19th-century scientist Michael Faraday. Born into a poor working-class family, he received little formal schooling but became interested in science while working as a bookbinder's apprentice. He is celebrated today for carrying out pioneering research into the relationship between electricity and magnetism. Faraday showed that if a wire was turned in the presence of a magnet or a magnet was turned in relation to a wire, an electric current was generated. This ground-breaking discovery led to the development of the electric generator and ultimately to modern power stations. During his life he became the most famous scientist in Britain and he played a key role in founding the Royal Institution's Christmas lectures which continue today.

With:

Geoffrey Cantor

Laura Herz

Frank James

Producer: Victoria Brignell.

Melvyn Bragg and his guests discuss the evolution and role of Circadian Rhythms, the so-called body clock that influences an organism's daily cycle of physical, behavioural and mental changes. The rhythms are generated within organisms and also in response to external stimuli, mainly light and darkness. They are found throughout the living world, from bacteria to plants, fungi to animals and, in humans, are noticed most clearly in sleep patterns.

With

Russell Foster

Debra Skene

And

Steve Jones

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

Timothy Gowers

And

Leslie Ann Goldberg

Producer: Simon Tillotson.

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

Frank Close

and

Steven Bramwell

Producer: Simon Tillotson.