The debate between Peletier and Clavius on superposition (Q1708441)

One of the more controversial aspects of the beginning of Euclid's \textit{Elements}, at least from the early modern period onward, is his use of superposition. For instance, in I.4, he demonstrates that two triangles with two pairs of equal sides and equal internal angles are equal to each other (SAS), using the fourth common notion (``things which coincide with one another equal each other''). This seems to rely on an apparently ungeometric notion, namely, superimposing one of the triangles on the other. Whether this is admissible was debated between Jacques Peletier du Mans and Christoph Clavius in the late 16th century, who had already sparred over the validity of the concept of an angle between a circle and its tangent line. The author of this paper analyzes both men's positions in detail. She argues that Peletier was not concerned with the kinematic implications of superposition, but rather made the case that superposition requires the judgment of the figures' congruence to be made based on intuition rather than reason. Clavius's response agreed with Peletier on the difficulty of the use of superposition in \textit{problems}, but defended its use in \textit{theorems} as a hypothetical (rather than constructive) process.