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Nonfiction 12

By Derek Zobel

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He published a translation of the Primi sei libri d’Euclide tratti in volgare (Milano, Lodovico Monza 1671). 45 This information can be obtained from the Ceva-Grandi correspondence quoted in the aforementioned article by Tenca, Relazioni fra Gerolamo Saccheri e il suo allievo Guido Grandi. E. Filippa, Euclidis priora elementa sex, auctore Carolo Eduardo Taurinense, reverendo patri Hieronymo Saccherio Societatis Iesu dicata, Torino, Zappata 1695. The fact that this work stresses the role of the Fifth Postulate in elementary geometry, though it is accepted as a postulate without attempting a proof, may be taken to attest to the atmosphere surrounding Saccheri’s teaching activity.

These are extensionally equivalent, but if the inference (→) is intended as an intensional operator expressing mathematical provability in classical sense (whatever this may mean), then the idea is that the passage (¬α→α) in consequentia mirabilis actually shows a proof of α. Thus, in concluding α through consequentia, we also have an explicit, direct proof of the statement, and therefore the conclusion is not indirect as in reductio arguments. At any rate, Saccheri regarded the consequentia mirabilis as a direct proof: cf.

55 Hence, we must try to determine how Saccheri intended to amend not only Clavius’ demonstration, but also its epistemological implications. It appears that the answer to this question can be found in the consequentia mirabilis. This is a deductive technique that had received a detailed theoretical examination in Saccheri’s Demonstrative Logic, and that he now applies in concreto in a couple of crucial propositions of Euclid Vindicated. A proof by consequentia mirabilis (which he calls redargutio, ‘redargution’) consists in deriving a proposition from its own negation, thereby establishing the unhypothetical validity of the proposition in question.

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