A mathematical instrument can sometimes be turned into a weapon, as will be argued here with the particular way Cardano and Ferrari used a compass in the context of the well-known polemic pitting Tartaglia against Cardano (1546-1548). Scholarship on Cardano’s mathematics is centred above all on diverse approaches to algebra; in this paper, however, an examination of the role of instruments in geometry offers a novel perspective on the divergence between the autodidact Tartaglia and his university-trained opponents. In fact, Tartaglia proposed the challenge of solving a number of Euclidean propositions by means of straight-edge and ‘fixed compass’ – that is, a compass set to a single, fixed opening. The use of such a fixed compass would likely have been most familiar to Tartaglia given his awareness of the techniques used by artisans and painters. In response to this challenge, Cardano and his pupil Ferrari showed how to reconstruct what they claimed to be all of Euclidean geometry (‘E così è provato tutto Euclide’ according to Ferrari) by invoking only a single fixed opening of the compass. Of course, their argument (in the Cartelli di matematica disfida and De subtilitate) offered a lot more mathematics than what was strictly ‘necessary’ for artisans. How should this type of answer be interpreted in a polemic context? In what way could it be seen as an attempt by Cardano and Ferrari to humiliate Tartaglia and deliberately show off their academic superiority? On the other hand, why did they label their work a ‘useless subtlety’? Apparently, neither Cardano nor his contemporaries were aware of having reached a very interesting result, at least seen from a retrospective mathematical point of view, because actually this research was appreciated anew only by 18th- and 19th-century geometers such as Mascheroni, Steiner and Poncelet.
The compass in transition: divergent views on a mathematical instrument in Cardano's and Tartaglia's circles
GAVAGNA, Veronica
2014
Abstract
A mathematical instrument can sometimes be turned into a weapon, as will be argued here with the particular way Cardano and Ferrari used a compass in the context of the well-known polemic pitting Tartaglia against Cardano (1546-1548). Scholarship on Cardano’s mathematics is centred above all on diverse approaches to algebra; in this paper, however, an examination of the role of instruments in geometry offers a novel perspective on the divergence between the autodidact Tartaglia and his university-trained opponents. In fact, Tartaglia proposed the challenge of solving a number of Euclidean propositions by means of straight-edge and ‘fixed compass’ – that is, a compass set to a single, fixed opening. The use of such a fixed compass would likely have been most familiar to Tartaglia given his awareness of the techniques used by artisans and painters. In response to this challenge, Cardano and his pupil Ferrari showed how to reconstruct what they claimed to be all of Euclidean geometry (‘E così è provato tutto Euclide’ according to Ferrari) by invoking only a single fixed opening of the compass. Of course, their argument (in the Cartelli di matematica disfida and De subtilitate) offered a lot more mathematics than what was strictly ‘necessary’ for artisans. How should this type of answer be interpreted in a polemic context? In what way could it be seen as an attempt by Cardano and Ferrari to humiliate Tartaglia and deliberately show off their academic superiority? On the other hand, why did they label their work a ‘useless subtlety’? Apparently, neither Cardano nor his contemporaries were aware of having reached a very interesting result, at least seen from a retrospective mathematical point of view, because actually this research was appreciated anew only by 18th- and 19th-century geometers such as Mascheroni, Steiner and Poncelet.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
