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The sources of the text and the various versions are described in great detail by Heiberg in the Prolegomena to Vol. III of his edition of Archimedes, where the editor supplements and, to some extent, amends what he had previously written on the same subject in his dissertation entitled Quaestiones Archimedeae (1879). It will therefore suffice here to state the main points of the discussion briefly.
The best manuscripts all share a common origin in a single manuscript that is no longer known to be in existence. It is described in one of the copies made from it (dating from between A.D. 1499 and 1531, which will be mentioned later) as "most ancient" (παλαιοτάτου), and all evidence suggests it was written as early as the 9th or 10th century. At one time, it was in the possession of George Valla, who taught at Venice between 1486 and 1499. Many important conclusions regarding its readings can be drawn from translations of parts of Archimedes and Eutocius made by Valla himself and published in his book titled De expetendis et fugiendis rebus (Venice, 1501). It appears to have been carefully copied from an original belonging to someone well-versed in mathematics, and it contained figures mostly drawn with great care and accuracy. However, there was considerable confusion between the letters used in the figures and those in the text. Following the death of Valla in 1499, this manuscript became the property of Albertus Pius Carpensis (Alberto Pio, Prince of Carpi). Part of his library passed through various hands and ultimately reached the Vatican, but the fate of the Valla manuscript seems to have been different; we hear of it being in the possession of Cardinal Rodolphus Pius (Rodolfo Pio), a nephew of Albertus, in 1544, after which it seems to have disappeared.