Historians have recently paid increasing attention to the role of the disputation in Italian universities and humanist circles. By contrast, the role of disputations as forms of entertainment at fifteenth-century Italian courts has been somewhat overlooked. In this article, the Milanese "scientific duel" (a courtly disputation) described in Luca Pacioli's De divina proportione is taken as a vantage point for the study of the dynamics of scientific patronage and social advancement as reflected in Renaissance courtly disputes. Pacioli names Leonardo da Vinci as one of the participants in the Milanese dispute. In this paper I argue that Leonardo's Paragone and Pacioli's De divina proportione are likewise the outcome of the Milanese "scientific duel." By challenging the traditional hierarchy of the arts, they both exemplify the dynamics of social and intellectual promotion of mathematicians and artists in the privileged setting of Renaissance courts, where courtly patronage could subvert the traditional disciplinary rankings.