Says Painter: “Seven years were devoted to the completion of the course in liberal arts [the Trivium and the Quadrivium].... Dialectic or logic was based somewhat remotely on the writings of Aristotle. At a later period, logic was rigidly applied to the development of theology, and gave rise to a class of scholars called the schoolmen.... Arithmetic was imperfectly taught, importance being attached to the supposed secret properties of numbers. Geometry was taught in an abridged form, while astronomy did not differ materially from astrology. The study of music consisted chiefly in learning to chant the hymns of the church.”[89] [Sidenote: Greater emphasis on logic] Mosheim thus continues his description of the work of the schools in the eleventh century: “This course of study, adopted in all the schools of the West, was not a little changed after the middle of this century. For logic, ... having been improved by the reflection and skill of certain close thinkers, and being taught more fully and acutely, acquired such an ascendency in the minds of the majority, that they neglected grammar, rhetoric, and the other sciences, both the elegant and the abstruse, and devoted their whole lives to dialectics, or to logical and metaphysical discussions. For whoever was well acquainted with dialectics, or what we call logic and metaphysics, was supposed to possess learning enough, and to lose nothing by being ignorant of all other branches of learning.... In this age, the philosophy of the Latins was confined wholly to what they called dialectics; and the other branches of philosophy were unknown even by name. Moreover their dialectics was miserably dry and barren.”[90] [Sidenote: Patristical Geography] This is sufficient, perhaps, on the use of language and logic, and we turn to geography and some of the sciences. Even the children to-day will smile at the teachings of some of the Church Fathers on the subject of geography. Says Draper: “In the Patristic Geography the earth is a flat surface bordered by the waters of the sea, on the yielding support of which rests the crystalline dome of the sky. These doctrines were for the most part supported by passages from the Holy Scriptures, perversely wrested from their proper meaning. Thus Cosmas Indicopleustes, whose Patristic Geography had been an authority for nearly eight hundred years, triumphantly disposed of the sphericity of the earth by demanding of its advocates, how, in the day of judgment, men on the other side of a globe could see the Lord descending through the air!”[91] [Sidenote: The beneficial work of explorers] It was in opposition to such theories, and a hundred absurdities concerning the ocean, the boiling waters of the equator, the serpents in the West, etc., that Columbus, De Gama, and other explorers had to contend; and one of the most wonderful effects of the work of these navigators was the thrust given papal education. A wound was then received which was incurable.