Melvyn Bragg and his guests discuss Euler's number, also known as e. First discovered in the seventeenth century by the Swiss mathematician Jacob Bernoulli when he was studying compound interest, e is now recognised as one of the most important and interesting numbers in mathematics. Roughly equal to 2.718, e is useful in studying many everyday situations, from personal savings to epidemics. It also features in Euler's Identity, sometimes described as the most beautiful equation ever written.

With:

Colva Roney-Dougal

June Barrow-Green

Vicky Neale

Producer: Thomas Morris.

Melvyn Bragg and his guests discuss the Sun. The object that gives the Earth its light and heat is a massive ball of gas and plasma 93 million miles away. Thanks to the nuclear fusion reactions taking place at its core, the Sun has been shining for four and a half billion years. Its structure, and the processes that keep it burning, have fascinated astronomers for centuries. After the invention of the telescope it became apparent that the Sun is not a placid, steadily shining body but is subject to periodic changes in its appearance and eruptions of dramatic violence, some of which can affect us here on Earth. Recent space missions have revealed fascinating new insights into our nearest star.

With:

Carolin Crawford

Yvonne Elsworth

Louise Harra

Producer: Thomas Morris.

Melvyn Bragg and his guests discuss the life and work of Robert Boyle, a pioneering scientist and a founder member of the Royal Society. Born in Ireland in 1627, Boyle was one of the first natural philosophers to conduct rigorous experiments, laid the foundations of modern chemistry and derived Boyle's Law, describing the physical properties of gases. In addition to his experimental work he left a substantial body of writings about philosophy and religion; his piety was one of the most important factors in his intellectual activities, prompting a celebrated dispute with his contemporary Thomas Hobbes.

With:

Simon Schaffer

Michael Hunter

Anna Marie Roos

Producer: Thomas Morris.

Melvyn Bragg and his guests discuss photosynthesis, the process by which green plants and many other organisms use sunlight to synthesise organic molecules. Photosynthesis arose very early in evolutionary history and has been a crucial driver of life on Earth. In addition to providing most of the food consumed by organisms on the planet, it is also responsible for maintaining atmospheric oxygen levels, and is thus almost certainly the most important chemical process ever discovered.

With:

Nick Lane

Sandra Knapp

John Allen

Producer: Thomas Morris

Melvyn Bragg and his guests discuss the science of matter and the states in which it can exist. Most people are familiar with the idea that a substance like water can exist in solid, liquid and gaseous forms. But as much as 99% of the matter in the universe is now believed to exist in a fourth state, plasma. Today scientists recognise a number of other exotic states or phases, such as glasses, gels and liquid crystals - many of them with useful properties that can be exploited.

With:

Andrea Sella

Athene Donald

Justin Wark

Producer: Thomas Morris.

Melvyn Bragg and his guests discuss the eye. Humans have been attempting to understand the workings and significance of the organ for at least 2500 years. Some ancient philosophers believed that the eye enabled creatures to see by emitting its own light. The function and structures of the eye became an area of particular interest to doctors in the Islamic Golden Age. In Renaissance Europe the work of thinkers including Kepler and Descartes revolutionised thinking about how the organ worked, but it took several hundred years for the eye to be thoroughly understood. Eyes have long attracted more than purely scientific interest, known even today as the 'windows on the soul'.

With:

Patricia Fara

William Ayliffe

Robert Iliffe

Producer: Thomas Morris.

Melvyn Bragg and his guests discuss Social Darwinism. After the publication of Charles Darwin's masterpiece On the Origin of Species in 1859, some thinkers argued that Darwin's ideas about evolution could also be applied to human society. One thinker particularly associated with this movement was Darwin's near-contemporary Herbert Spencer, who coined the phrase 'survival of the fittest'. He argued that competition among humans was beneficial, because it ensured that only the healthiest and most intelligent individuals would succeed. Social Darwinism remained influential for several generations, although its association with eugenics and later adoption as an ideological position by Fascist regimes ensured its eventual downfall from intellectual respectability.

With:

Adam Kuper

Gregory Radick

Charlotte Sleigh

Producer: Thomas Morris.

Melvyn Bragg and his guests discuss Catastrophism, the idea that natural disasters have had a significant influence in moulding the Earth's geological features. In 1822 William Buckland, the first reader of Geology at the University of Oxford, published his famous Reliquae Diluvianae, in which he ascribed most of the fossil record to the effects of Noah's flood. Charles Lyell in his Principles of Geology challenged these writings, arguing that geological change was slow and gradual, and that the processes responsible could still be seen at work today - a school of thought known as Uniformitarianism. But in the 1970s the idea that natural catastrophes were a major factor in the Earth's geology was revived and given new respectability by the discovery of evidence of a gigantic asteroid impact 65 million years ago, believed by many to have resulted in the extinction of the dinosaurs.

With:

Andrew Scott

Jan Zalasiewicz

Leucha Veneer

Producer: Thomas Morris.

Melvyn Bragg and his guests discuss complexity and how it can help us understand the world around us. When living beings come together and act in a group, they do so in complicated and unpredictable ways: societies often behave very differently from the individuals within them. Complexity was a phenomenon little understood a generation ago, but research into complex systems now has important applications in many different fields, from biology to political science. Today it is being used to explain how birds flock, to predict traffic flow in cities and to study the spread of diseases.

With:

Ian Stewart

Jeff Johnson

Professor Eve Mitleton-Kelly

Producer: Thomas Morris.

Melvyn Bragg and his guests discuss the development of the microscope, an instrument which has revolutionised our knowledge of the world and the organisms that inhabit it. In the seventeenth century the pioneering work of two scientists, the Dutchman Antonie van Leeuwenhoek and Robert Hooke in England, revealed the teeming microscopic world that exists at scales beyond the capabilities of the naked eye.

The microscope became an essential component of scientific enquiry by the nineteenth century, but in the 1930s a German physicist, Ernst Ruska, discovered that by using a beam of electrons he could view structures much tinier than was possible using visible light. Today light and electron microscopy are among the most powerful tools at the disposal of modern science, and new techniques are still being developed.

With:

Jim Bennett

Sir Colin Humphreys

Michelle Peckham

Producer: Thomas Morris.