In proposition 3 of the κύκλου μέτρησις Archimedes constructs two rational numbers that approximate the measure of the ratio between the circumference and its diameter, today known as π. In this work we emphasize two aspects of Archimedes' proof. The first one concerns the choices of the rational values approximated by excess and defect of √3 used at the beginning of the calculation procedure. These values turn out to be the best approximants based on the modern theory of continued fractions, whose terms can be obtained from the ancient Euclidean ανθυφαίρεσις. The second one concerns some estimates used by Archimedes, apparently unjustified and sub-optimal, which instead are the result of a precise logical procedure that we recontruct in terms of an algorithm. We also propose an educational transposition of this study that starts from the ancient text and leads to the production of a commented translation in which the student is the protagonist of the discovery process.
Nella proposizione 3 della κύκλου μέτρησις Archimede costruisce due numeri razionali che approssimano la misura del rapporto tra la circonferenza e il suo diametro, oggi noto come π. In questo lavoro vogliamo porre l’accento su due aspetti del trattato di Archimede: il primo riguarda le scelte dei valori razionali approssimati per eccesso e per difetto di √𝟑 utilizzati all’inizio della procedura di calcolo, che risultano essere i migliori approssimanti in base alla moderna teoria delle frazioni continue, i cui termini si possono ricavare dall’antico ανθυφαίρεσις euclideo; il secondo riguarda alcune stime utilizzate da Archimede, apparentemente non giustificate e sub-ottime, che invece sono frutto di un preciso procedimento logico da noi ricostruito in termini di un algoritmo. Di tale studio è stata proposta una trasposizione didattica che parte dal testo antico e giunge alla produzione di una traduzione commentata in cui lo studente è protagonista del processo di scoperta.
Proposizione 3 della κύκλου μέτρησις: la più emblematica delle opere di Archimede (Proposition 3 of the κύκλου μέτρησις: the most emblematic of Archimedes' works) / Palma, Antonella; Dragone, Luca. - In: SCIENCE & PHILOSOPHY. - ISSN 2282-7765. - 12:1 2024(2024), pp. 279-287. [10.23756/sp.v12i1.1628]
Proposizione 3 della κύκλου μέτρησις: la più emblematica delle opere di Archimede (Proposition 3 of the κύκλου μέτρησις: the most emblematic of Archimedes' works)
Palma, Antonella;Dragone, Luca
2024
Abstract
In proposition 3 of the κύκλου μέτρησις Archimedes constructs two rational numbers that approximate the measure of the ratio between the circumference and its diameter, today known as π. In this work we emphasize two aspects of Archimedes' proof. The first one concerns the choices of the rational values approximated by excess and defect of √3 used at the beginning of the calculation procedure. These values turn out to be the best approximants based on the modern theory of continued fractions, whose terms can be obtained from the ancient Euclidean ανθυφαίρεσις. The second one concerns some estimates used by Archimedes, apparently unjustified and sub-optimal, which instead are the result of a precise logical procedure that we recontruct in terms of an algorithm. We also propose an educational transposition of this study that starts from the ancient text and leads to the production of a commented translation in which the student is the protagonist of the discovery process.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.