Reading the "General Trattato" of Niccolò Tartaglia as a textbook of geometry

Niccolò Tartaglia (1499-1557) spent most of his life in Venice, one of the cradles of Italian humanism but also a town where lots of merchants, engineers and craftsmen worked. He was an interlocutor of both the intellectual and practical environment, and this is the reason why his work is so rich of hints and it is so representative of the mathematical Renaissance. Tartaglia did not write a textbook based on his experience of teaching, nevertheless we can consider the General Trattato (1557-1560) the mirror of his double soul: abacus teacher and restorer of Greek mathematics (Euclid and Archimedes mainly). The First Part, in fact, is a sort of abacus treatise where is not too difficult detecting the experience due to the arithmetical teaching. Now it is time to investigate, from this viewpoint, also the teaching of geometry. The proposed talk will be focused essentially on the Third, Fourth and Fifth Part of General Trattato, that concern geometry. To be precise, the Third and the Fourth Part are mainly devoted to a reworking of Euclidean and Archimedean geometry from a practical point of view and to measurement problems; the Fifth Part is addressed to ‘geometers, architects, engineers…’ and is devoted to construction problems (also with a rule and a fixed compass). These Parts are to be considered as an essential complement of Tartaglia’s translation of the Elements (1543) and actually several of the “translator’s comments” explicitly refer to General Trattato. But in the same Parts we can collect a number of clues related to the teaching of geometry: their description and contextualization is the aim of the present contribution.