Late 19th-century maps of the solar system showed a planet between Mercury and the Sun that never existed. This episode is the biography of Vulcan, a world willed into being by mathematics. It begins with Urbain Le Verrier, the man who had already discovered Neptune at the tip of his pen, finding in 1859 that Mercury's perihelion advanced by 43 arcseconds per century more than Newton's laws allowed, and concluding that an unseen mass must be responsible.

The episode follows the 1859 report of country doctor Edmond Lescarbault, who timed a black dot crossing the Sun with a pendulum he used to take patients' pulses, Le Verrier's unannounced visit to verify it, and the Legion of Honour that followed, while the astronomer Emmanuel Liais, watching with a better telescope at the same time, saw nothing. It recounts the 1878 eclipse claims of James ...