into 27 parts with 9 number symbols (this would exclude zero, which was represented by an empty column) and 1,000 characters in all, crafted out of animal horn by a shieldmaker of Rheims. According to his pupil Richer, Gerbert could perform speedy calculations with his abacus that were extremely difficult for people in his day to think through using only Roman numerals. Due to Gerbert's reintroduction, the abacus became widely used in Europe once again during the 11th century. Armillary sphere and sighting tube Although lost to Europe since the terminus of the Greco-Roman era, Gerbert reintroduced the astronomical armillary sphere to Latin Europe via the Islamic civilization of Al-Andalus, which was at that time at the "cutting edge" of civilization. The details of Gerbert's armillary sphere are revealed in letters from Gerbert to his former student and monk Remi of Trèves and to his colleague Constantine, the abbot of Micy, as well as the accounts of his former student and French nobleman Richer, who served as a monk in Rheims. Richer stated that Gerbert discovered that stars coursed in an oblique direction across the night sky. Richer described Gerbert's use of the armillary sphere as a visual aid for teaching mathematics and astronomy in the classroom. Historian Oscar G. Darlington asserts that Gerbert's division by 60 degrees instead of 360 allowed the lateral lines of his sphere to equal to six degrees. By this account, the polar circle on Gerbert's sphere was located at 54 degrees, several degrees off from the actual 66° 33'. His positioning of the Tropic of Cancer at 24 degree was nearly exact, while his positioning of the equator was correct by definition. Richer also revealed how Gerbert made the planets more easily observable in his armillary sphere: He succeeded equally in showing the paths of the planets when they come near or withdraw from the earth. He fashioned first an armillary sphere. He joined the two circles called by the Greeks coluri and by the Latins incidentes because they fell upon each other, and at their extremities he placed the poles. He drew with great art and accuracy, across the colures, five other circles called parallels, which, from one pole to the other, divided the half of the sphere into thirty parts. He put six of these thirty parts of the half-sphere between the pole and the first circle; five between the first and