The “projective spirit” in Segre’s lectures on differential equations.

The “projective spirit” was undoubtedly a characteristic of Italian research in mathematics in the period 1860-1940, which pervades the successful researches of Luigi Cremona in projective geometry and Corrado Segre, Guido Castelnuovo, Federigo Enriques and Francesco Severi in algebraic geometry. However, its un- conditional pursuit was also one of the reasons of the eclipse of Italian research in algebraic geometry in the period between the two world wars. This ”projective spirit” was not limited to algebraic geometry but it can be traced, at various degree, in the background and sensibility of many Italian mathematicians of that period. It was an essential part of the mathematical culture disseminated at Italian Universities. The course given by Corrado Segre in the academic year 1920–1921 on the geometric theory of differential equations is, in some sense, paradigmatic of the pervasiveness of this spirit.