Liberal arts does not mean "humanities" as opposed to "sciences." The term, for one thing, long predates the entry of "science" into the western vocabulary. The artes liberales, where ars means "skill" (as is usual; Hogwarts isn't teaching students Defense Against Dark Paintings and Statuary), were and are a series of seven disciplines subdivided into two groups, one of three (the trivium) and one of four (the quadrivium). They are called liberales because they are the skills required for a free person to be considered educated. The first reference to them is in Cicero, de oratore 1.16: [artes] quae libero dignae, "the arts which befit a free person." The liberal arts were entirely intellectual, requiring no manual labor, unlike the artes serviles or vulgares, essentially trades and crafts. Originating in Greek thought and passed to medieval society by the Romans, the artes liberales consisted of rhetoric, grammar, arithmetic, logic, astronomy, music, and geometry. These were further subdivided into the trivium (grammar, logic, rhetoric) and the quadrivium (arithmetic, astronomy, music, geometry). The intellectual path from the quadrivium in particular to modern mathematics and physics is not a hard one to work out, but its historical development should be outlined. Already by the later Middle Ages Thomas Aquinas was noting that the trivium and quadrivium together were insufficient as a framework of education regarding certain philosophical studies. The artes liberales, always conceived of as a set unit that must be taught together, were by the late Middle Ages being broken up into independent disciplines. The trivium was more neatly organized--all three of its subjects dealt with language--and held out somewhat better, and instead of being preliminary to the quadrivium was transferred into the universities alongside it, especially by the humanists. The Renaissance saw changes to the framework even of the trivium, however: grammar became more generally philology, and added were poetry, moralizing philosophy, and history. The quadrivium had already been dying in the Renaissance, but to the trivium humanist scholars added physics and metaphysics.