Foundations of Geometry in the School of Peano

Abstract

The question that motivated this paper — why Pieri made an analogy to Peano’s affinities when he introduced segmental transformations — revealed several plausible answers. But perhaps more importantly, seeking to answer the question provided an opportunity to explore the commonalities and differences about the scholars’ views and treatments of projective geometry and its transformations. In this regard, there is one more avenue to explore. What is not evident from his axiomatizations, but is clear from his lectures37 to students, is the evolution of Pieri’s thoughts about projective geometry. In his (1891) notes for a course in projective geometry at the Military Academy — prior to writing his first axiomatization, but after he had translated Staudt (1847) — Pieri took the same approach to projective geometry as had Peano38. But in his notes for the University of Parma (1909–10), after he had written all his foundational papers in projective geometry, Pieri alerted students to the more “desirable” direction of Staudt as opposed to that pursued by J. Poncelet, Mobius, J. Steiner and Chasles, who studied projective geometry as an extension of elementary geometry.