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 many more equally difficult theorems might be quoted which are original with Pappus as far as we know. It ought to be remarked, however, that he is known in three instances to have copied theorems without giving due credit, and that he may have done the same thing in other cases in which we have no data by which to ascertain the real discoverer.

About the time of Pappus lived Theon of Alexandria. He brought out an edition of Euclid's Elements with notes, which he probably used as a text-book in his classes. His commentary on the Almagest is valuable for the many historical notices, and especially for the specimens of Greek arithmetic which it contains. Theon's daughter Hypatia, a woman celebrated for her beauty and modesty, was the last Alexandrian teacher of reputation, and is said to have been an abler philosopher and mathematician than her father. Her notes on the works of Diophantus and Apollonius have been lost. Her tragic death in 415 is vividly described in Kingsley's Hypatia.

From now on, mathematics ceased to be cultivated in Alexandria. The leading subject of men's thoughts was Christian theology. Paganism disappeared, and with it pagan learning. The Neo-Platonic school at Athens struggled on a century longer. Proclus, Isidorus, and others kept up the "golden chain of Platonic succession." Proclus, the successor of Syrianus, at the Athenian school, wrote a commentary on Euclid's Elements. We possess only that on the first book, which is valuable for the information it contains on the history of geometry. Damascius of Damascus, the pupil of Isidorus, is now believed to be the author of the fifteenth book of Euclid. Another pupil of Isidorus was Eutocius of Ascalon, the commentator of Apollonius and Archimedes. Simplicius wrote a commentary on Aristotle's De Cœlo. In the year 529, Justinian, disapproving heathen learning, finally closed by imperial edict the schools at Athens.