The Discourses and Mathematical Demonstrations Relating to Two New Sciences (Discorsi e dimostrazioni matematiche intorno a due nuove scienze, 1638) was Galileo's  final book and a sort of scientific testament covering much of his work in physics over the preceding thirty years.

After his Dialogue Concerning the Two Chief World Systems, the Roman Inquisition had banned publication of any work by Galileo, including any he might write in the future. After the failure of attempts to publish the work in France, Germany, or Poland, it was picked up by Lodewijk Elzevir in Leiden, The Netherlands, where the writ of the Inquisition was of little account (see House of Elzevir).

The same three men as in the Dialogue carry on the discussion, but they have changed. Simplicio, in particular, is no longer the stubborn and rather dense Aristotelian; to some extent he represents the thinking of Galileo's early years, as Sagredo represents his middle period. Salviati remains the spokesman for Galileo.