Transmission of the Greek Classics on music ("De institutione musica libri quinque", unfinished) contributed to medieval education. "De arithmetica", begins with modular arithmetic, such as even and odd, evenly even, evenly odd, and oddly even. He then turns to unpredicted complexity by categorizing numbers and parts of numbers. His translations of Euclid on geometry and Ptolemy on astronomy, if they were completed, no longer survive. Boethius made Latin translations of Aristotle's "De interpretation" and "Categories" with commentaries. These were widely used during the Middle Ages. In the Western Provinces (what today is considered the Western Europe's heartland), the collapsing Roman empire lost many Greek manuscripts
Ptolemy I of Tusculum (temporarily) resumed. Ptolemy and his nephew the prefect were holding the Castel Sant'Angelo. Peter Colonna and Rainald Senebaldi turned to the pope's side and, soon after Christmas, Ptolemy and the rest of the imperialists in the Eternal City were forced to flee by the pope, who soon died. Ptolemy I of Tusculum Ptolemy I ( or ; died 1126) was the count of Tusculum in the first quarter of the twelfth century. He was a son of Gregory III. Peter Pisanus, in his "Vita Paschalis II" refers to Ptolemy and the abbot of Farfa as the allies of the emperor