Richer also speaks of Gerbert’s epoch-making mathematical knowledge.[367] In arithmetic he improved the current methods of computation; in geometry he taught the traditional methods of measurement descended from the Roman surveyors, and compiled a work from Boëthius and other sources. For astronomy he made spheres and other instruments, and in music his teaching was the best obtainable. In none of these provinces was he an original inventor; nor did he exhaust the knowledge had by men before him. He was, however, the embodiment of mediaeval progress, in that he drew intelligently upon the sources within his reach, and then taught with understanding and enthusiasm. Richer’s praise is unstinted: “He began with arithmetic; then taught music, of which there had long been ignorance in Gaul.... With what pains he set forth the method of astronomy, it may be well to state, so that the reader may perceive the sagacity and skill of this great man. This difficult subject he explained by means of admirable instruments. First he illustrated the world’s sphere by one of solid wood, the greater by the less. He fixed it obliquely as to the horizon with two poles, and near the upper pole set the northern constellations, and by the lower one those of the south. He determined its position by means of the circle called by the Greeks _orizon_ and by the Latins _limitans_, because it divides the constellations which are seen from those which are not. By his sphere thus fixed, he demonstrated the rising and setting of the stars, and taught his disciples to recognize them. And at night he followed their courses and marked the place of their rising and setting upon the different regions of his model.” The historian passes on to tell how Gerbert with ingenious devices showed on his sphere the imaginary circles called parallels, and on another the movements of the planets, and on still another marked the constellations of the heavens, so that even a beginner, upon having one constellation pointed out, could find the others.[368]