The foundation of differential calculus was a pivotal topic of discussion since it was taught at the Ecole Polytechnique from the end of the 18th century onwards. In this presentation, I will examine a variety of sources, including published lectures, manuscripts and textbooks, with the aim of elucidating the practices of a group of teachers in this institution who were involved with the process of rigorizing analysis, including Lagrange and Fourier. The practice of teaching provided both Lagrange and Fourier with the opportunity to address the issue with rigor and clarity, particularly when presenting the subject to students who lacked familiarity with this branch of mathematics. Both Lagrange and Fourier played a role in the formation of a mathematical culture that placed value on rigor as an epistemological value. They did so by advocating for and demanding rigorous treatment of the subject. Cauchy may well have been influenced by this mathematical culture, given that he studied and taught later at the Ecole Polytechnique, where this culture had been formed. However, a further investigation reveals that they differ in practice with regard to some core issues in analysis, including the use of geometry in analysis and the attitude toward infinitesimals. This demonstrates that practices can vary even within a similar mathematical culture.

While knowledge of ancient Greek was lost in the Middle Ages in the West, this was not the case in the Byzantine Empire during the same period. Ancient mathematical treatises continued to be read in Greek. However, mathematical and scientific knowledge evolves between late Antiquity and the Middle Ages, so that relative linguistic continuity does not guarantee cultural continuity between Hellenistic Antiquity and the Byzantine Middle Ages.

In fact, we postulate the opposite. To illustrate this, we will use case studies drawn from material sources from this period, i.e. manuscripts copied in the Middle Ages. We will compare the medieval practice of studying various books of Euclid's Elements with that of ancient treatises on cosmology and astronomy, based on the scholia and marginal annotations of these treatises, which can be associated with a Byzantine cultural context.

The study of the treatment of celestial dimensions in Arabic astronomical texts written before the middle of the eleventh century documents the existence of a great diversity of conceptual traditions within what is too often referred to as a unified whole called ‘Arab science’ or ‘Islamic science’. I will also document the persistence and coexistence of some of these traditions from Greek antiquity to the medieval Latin period.