11. Transmitting Knowledge; The Quadrivium: Arithmetic, Geometry, Astronomy, and Music
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The second set of subjects that, together with the trivium, formed the liberal arts comprised arithmetic, geometry, astronomy, and music. The quadrivium maintained its essential medieval formulation well into the early modern period, even as new astronomical discoveries slowly chipped away at old certainties. Music theory, for example, was profoundly marked by Boethius’s diagrammatic formulations for a millennium or more. In the same period, new practical applications for traditional mathematical knowledge proliferated in fields as diverse as banking, international trade, and seafaring, ensuring a growing market for new treatises on these subjects. In the field of astronomy, the geocentric world system of Claudius Ptolemy (ca. 100–ca.170) remained the norm throughout the Mediterranean world. However, it was not until the collaboration of Georg von Peuerbach (1423–61) and Regiomontanus (1436–76), that more precise calculations of the movements of planets, stars, and comets through the heavens were produced. The complexity of the rotating volvelles required for proper comprehension of these astronomical models prompted the production of manuscript versions of these aids, to be used alongside the printed texts. At the same time, practitioners of art forms traditionally classified as mechanical began to assert themselves intellectually on the basis of their substantial engagement with the quadrivium. The art of architecture found its greatest proponent in Leon Battista Alberti (1404–72), who also penned a sophisticated treatise that proclaimed a higher social status for the painter, a notion seconded by such prominent literary figures as Baldassare Castiglione (1478–1529).