Passing to the Quadrivium, we find that Fulbert had studied its four branches under Gerbert. In Arithmetic the students used the treatise of Boëthius, and also the Abacus, a table of vertical columns, with Roman numerals at the top to indicate the order of units, tens, and hundreds according to the decimal system. In Geometry the students likewise fell back upon Boëthius. Astronomy, the third branch of the Quadrivium, had for its practical object the computation of the Church’s calendar. The pupils learned the signs of the Zodiac and were instructed in the method of finding the stars by the _Astrolabius_, a sphere (such as Gerbert had constructed) representing the constellations, and turning upon a tube as an axis, which served to fix the polar star. Music, the fourth branch of the Quadrivium, was zealously cultivated. For its theory, the treatise of Boëthius was studied; and Fulbert and his scholars did much to advance the music of the liturgy, composing texts and airs for organ chanting. In addition to the Quadrivium, medicine was taught. The students learned receipts and processes handed down by tradition and commonly ascribed to Hippocrates. For more convenient memorizing, Fulbert cast them into verse. Such “medicine” was not founded on observation; and a mediaeval scholar-copyist would as naturally transcribe a medical receipt-book as any other work coming within the range of his stylus. One may remember that in the early Middle Ages the relic was the common means of cure. The seven _Artes_ of the Trivium and Quadrivium were the handmaids of Theology; and Fulbert gave elaborate instruction in this Christian queen of the sciences, expounding the Scriptures, explaining the Liturgy, and taking up the controversies of the time. As a part of this sacred science, the students apparently were taught something of Canon and Roman law and of Charlemagne’s Capitularies.[381] IV The Chartres Quadrivium represents the extreme compass of mathematical and physical studies in France in the eleventh century, when slight interest was taken in physical science--a phrase far too grand to designate the crass traditional views of nature which prevailed. Indifference to natural knowledge was the most palpable intellectual defect of Ambrose and Augustine, and the most portentous. The coming centuries, which were to look upon their writings as universal guides to living and knowing, found therein no incentive to observe or study the natural world. Of course the Carolingian period evolved out of itself no such desire; nor did the eleventh century. At the best, the general understanding of physical fact remained that which had been handed down. It was gleaned from the books commonly read, the _Physiologus_ or the edifying stories of miracles in the myriad _Vitae Sanctorum_, quite as much as from the scant information given in Isidore’s _Origines_, Bede’s _Liber de temporibus_, or the _De universo_ of Rabanus Maurus.