The Reception by Peano of Pasch's Vorlesungen über neuere Geometrie Peano wrote the Calcolo geometrico in 1888. One year later, in 1889, he published I principii di geometria , in which he developed, in Pasch's wake, an axiomatisation of geometry. What is the relationship between this work and the previous elaboration of a geometrical calculus ? In this article, we outline the deep difference between the two methods : although the construction of a geometrical algebra aimed at showing the specificity of spatial magnitudes, the axiomatisation does not refer to the ontological distinction between geometrical and numerical entities. However, we show that the way Peano read Pasch's Vorlesungen depended on his previous involvement in the Grasmannian tradition : for him, the segment AB does not refer (as it did for Pasch) to an observable object, rather, AB designates the result of a new geometrical product. In the end, the situation is quite complicated : on the one hand, the algebra and the axiomatic are grounded, in Peano's thought, on completely opposite conceptions of geometrical objects ; on the other, the axiomatic method, as it is developed in I principii , results directly from an interpretation of Pasch entirely based on the Calcolo .
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