... Ascoli does not limit himself to dryly following the guidelines for university courses, but attempts to "discern in the admirable edifice of concepts and results that small bit that is essential in the very first study from what would otherwise be destined to remain a lifeless and inexpressive knowledge; to present to young people a simple and harmonic organism of fundamental ideas that contribute effectively towards their intellectual formation" (Preface). This is the reason why he introduces the concept of the integral before that of the derivative, because it is simpler and more intuitive when the point of departure is the calculation of areas.

Coming to Turin - where, in 1925, he also married - was undoubtedly a decisive event in the life and career of Ascoli. Indeed, at that time his activity was not - as can be seen in the dozen works published up to that time - significantly different from what one would expect of good teacher, of an excellent middle school teacher. However, in 1926, on the threshold of becoming forty years old, there was an abrupt change that seems to justify the title of a famous book: "Life begins at forty".

As for me, in October I was upset by being given a lot of work that I had never expected, to which was added in November, private tutoring of superior character that was offered to me by the Jewish Community in Milan and for which I went there three times a week, with much loss of time; my occupation for the rest of the time is not unpleasant and so I gave my consent to it from September. On the other hand there are times when each of my skills is put to good use.

In those times, when so many relationships were inevitably lost or destroyed, your constant remembrance and your faithful friendship, which remained faithful even in the worst of times - and I want to proclaim this at every opportunity, to honour your courage and your right-mindedness - were a great comfort to me. Many times in recent years, putting pressure on my natural optimism, there has been created in me a great contempt for men, and especially those of high standing and ingenuity who should also be an example of honesty and character. The participation in the purge of the University of Milan, put me in contact with so many small and large acts of cowardice - it was the final blow for me. To be able to find among the ex-fascist university a good person, who after the end of fascism, has only served to put in place a good and beautiful idea that has done and is doing great honour to Italy, outside and above that of sectarianism, is a real consolation.

... professional development course aimed at new graduates who had to participate in the competitions for teaching positions in secondary schools. The course lasted three hours a week and comprised lessons in mathematics aimed at completing or refreshing the basic preparation and to accustoming future teachers to look at elementary mathematics from an advanced standpoint, as well as to take into consideration the history of mathematics; these lessons were flanked by exercises and discussion on the topics that would be addressed in the competitions. These activities of Ascoli came together in the book 'Lezioni di Matematiche complementari' Ⓣ (1952, second edition 1954) and in the volumes on the topics assigned in the course on mathematical culture for new graduates, 'Svolgimento dei temi assegnati nel corso di cultura matematica dell'Università di Torino' Ⓣ (1953, 1955, 1957).

On the one hand he was always attentive to the latest developments in analysis while, on the other hand, he had his roots in the classic treatment of the great masters, and found nourishment in the history of mathematics, in logic, and in methodology. He considered the great problems of science, that is, what is the place and the value of mathematics in general, and of analysis in particular.