Algebra: The Eighth Liberal Art?

Abstract

What is the role of algebra in classical mathematics education? How does it relate to the four quadrivial arts? These questions have troubled the mathematical community since the introduction of algebra into the Renaissance academy by men like Fran\c{c}ois Vi\`ete, Guillame Gosselin, and Ren\'e Descartes. Their challenge is perhaps most starkly articulated at the conclusion of Vi\`ete's Introduction to the Analytic Art, where he claims that his algebra "appropriates to itself by right the proud problem of problems, which is: [sic] TO LEAVE NO PROBLEM UNSOLVED". Some contemporary educators respond by eschewing these methods to avoid the excessive formalization often accompanying algebra, and to give a central place to the geometrical tradition of Euclid's Elements. Others embrace the rise of algebra in the curriculum, focusing on contemporary techniques and priorities. This paper seeks to reconcile these perspectives by clarifying the way in which algebra participates in the quadrivial arts. Based on testimony from both the origins of algebra and its contemporary practitioners, I argue that algebra is not so much an eighth liberal art as an arithmetical language of form; an actualized potential in arithmetic. I conclude by offering curricular recommendations which provide glimpses of the practical insights available from this vantage.