From two documents published by Professor Gherardi,[570] we learn that in 1639 Galileo once more asked at Rome for some favours not specified, but that they were absolutely refused by the Pope. From this time Galileo came no further into direct contact with the Roman curia. He had been compelled to give up all hope of any amelioration of his lot from the implacable Urban VIII. So he ended his days quietly and resigned, as the prisoner of the Inquisition, in his villa at Arcetri. Castelli also, who (as his letters to Galileo of 1639 bear witness)[571] had warmly exerted himself on his behalf with Cardinal Barberini and other influential persons, had probably come to the conclusion that nothing more could be done for his unfortunate friend, for from this time we find nothing in his letters to Galileo but scientific disquisitions and spiritual consolations.[572] This indicates the two interests which occupied the latest period of Galileo’s life—deep piety and scientific meditations. His utter hopelessness and pious resignation are very clearly expressed in the brief sentence he used often to write to Castelli: “Piace cosi a Dio, dere piacere cosi ancora a Noi.”[573] (If it please God, it ought also to please us.) He never omitted in any letter to his old friend and pupil to commend himself in conclusion to his prayers,[574] and in his letter of 3rd December, 1639, he added: “I remind you to persevere in your prayers to the all-merciful and loving God, that He will cast out the bitter hatred from the hearts of my malicious and unhappy persecutors.” The lofty genius with which nature had endowed Galileo never displayed itself in so striking and surprising a manner as during these last three years. No sooner were his physical sufferings in some measure relieved, than he occupied himself in scientific speculations, the results of which he partly communicated to his great pupil and subsequent biographer, Viviani, by word of mouth, and partly dictated them to some of those about him. The society of young Viviani, then eighteen years of age, who, by permission of the Inquisition, spent the last two years and a half of the old master’s life near him,[575] was the greatest comfort to him, and he conceived a fatherly affection for the talented youth. We owe it partly to the assistance and stimulus given by Viviani that the aged Galileo worked on to the end in improving and enlarging his “Dialoghi delle Nuove Scienze,” made a number of additions, and added new evidence of great importance to science in two supplementary dialogues.[576]