In 1609, Galileo was in Venice to show the Doge his telescope when he encountered the First Doctor, Steven Taylor, and Vicki Pallister, who had come to Venice responding to a mysterious invitation. Since the Doctor required Galileo's assistance at the same time he was scheduled to demonstrate the invention, Steven had to impersonate Galileo by means of a holographic disguise. The Doctor also helped Galileo improve his telescope design. (PROSE: The Empire of Glass)
He told him he should remain quiet about some of his ideas that would be seen as heretical. (AUDIO: The Galileo Trap)
The Eighth Doctor met Galileo later in his life, after he had lost his sight. (AUDIO: The Galileo Trap) Galileo later saw the effects of the Eleven's stellar manipulator on the sun, believing it to be the end of the world. (AUDIO: The Satanic Mill)
The only complete transcript of Galileo Galilei's trial was kept in the Library of St John the Beheaded. (PROSE: All-Consuming Fire)
References 
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Professor J. P. Kettlewell mentioned Galileo while speaking about unappreciated scientists with the Fourth Doctor. (TV: Robot)
In the late 15th century, the Fourth Doctor noted, while using Giuliano's telescope, that in another fifty years they could have used Galileo's. (TV: The Masque of Mandragora)
The Fourth Doctor thought Galileo would be impressed by in-station fusion satellites, devices used as artificial suns. (TV: The Sun Makers)
Sarah Jane Smith said that she didn't think that Galileo and Copernicus were carrying out experiments in the paranormal. (TV: The Lost Boy)
