Born in Turin in 1889, Alessandro Terracini grew up at the School of D’Ovidio, Fano, Fubini, and Segre (the latter acting as supervisor of his degree thesis), all of whom inspired his research in projective-differential geometry and left a lasting impression on him. The quintessentially Italian style of that School to which Terracini would always claim to belong emerges both in his early research on tangent and osculating spaces to a manifold, in relation to their possible singularities, and in his later works on the geometric meaning of the projective normal, W-congruences and the concept of approximation order in the incidence of two planes or spaces. It is equally apparent in the essay Esposizione di alcuni risultati di geometria proiettiva differenziale negli iperspazi, published as an Appendix to the treatise by Fubini and Čech, Geometria proiettivo-differenziale (Bologna, Zanichelli, II, 1927, pp. 729–69), in which, despite using differential equation systems, Terracini accompanied each analytical result with a clear geometric interpretation in order to demonstrate how analytical and synthetic procedures can be integrated and illuminate each other.