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 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.

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: "Place cosi a Dio, dere piacere cosi ancora a Noi." (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, 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, 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