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Original scientific paper

Cavalieri, Fabri and Gradić on Galileo's Paradox of the Bowl

Ivica Martinović

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Two scientists who stand out among all the mathematicians who evaluated Galileo’s paradox of the bowl i.e. the paradox of equality of point and line, before Gradić are Bonaventura Cavalieri, Gradić’s professor of mathematics in Bologna and Honoré Fabri, Gradić’s correspondent in 1661 the year Gradić wrote De loco Galilei. Cavalieri utilized the proof of equality between volumes of cone and crater of leading to Galileo’s paradox in his famous work Geometria indivisibilibus continuorum and formulated objections arisen from his understanding of the method of indivisibles in the correspondence with Galileo in 1634. Fabri’s viewpoint on Galileo’s paradox was included in his lectures of physics Tractatus physicus de motu locali (1646). He consistently applied the method of indivisibles and referred on detailed classification of the problem compositio et resolutio continui in his metaphysics. The comparative study of Galileo’s paradox by Cavalieri and Fabri enables the reevaluation of Gradić’s treatise. Opposed to Fabrio, Gradić insisted on mathematical argumentation. He introduced the concept of the uniformis processio of a plane in order to describe the approaching to the limit of a geometric creation. Gradić’s treatise contains the original description of transition from the finite to the infinite number of portions of cone and cater, i.e. form the denticulated forms to the perfect solids of cone and crater. This is confirmed by the rigor of mathematical statements and by the concept of whatever approximation quaecumque proximitas. Researching the relation between geometric object and its limit Gradić pointed out that the starting point of Cavalieri’s method of indivisibles »limit is not component part of geometric object«, kept from verifying Euclid’s third axiom on the equality of residues. Studying the limiting process toward the infinitesimal method Gradić offered an authentic way, which was not recognized in the 17th century and after.


Bonaventura Cavalieri, Honoré Fabri, Stjepan Gradić, Galileo

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