De quelques questions touchant au traitement de la proportionnalité dans les Eléments d'Euclide
Institution:
Paris, EHESSDisciplines:
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Abstract EN:
This work is organized in four parts. The first is hostoriographic; several important problems are still unsolved : the nature of the basic objects of the theory of prioportions (ratio or proportions, numbers and or magnitudes), the history of this theory before and after euclid, the influence on greek mathematicians of the hellenistic and roman periods. The first problem is examined in the second and third parts; the second part deals with fundamental objets : numbers and magnitudes and the relations which can hold between them. Two notions are of special importance : multiple (plethos) and equimultiplicity. On this basis it isquite possible to explain and justify the double euclidean treatment of proportionality, considered by most commentators as a real difficulty of the elements' sturcture. The question of the effective thermatization of "ratio" and "proportion" is taken up anew in the third part, by examining how these notions are used in the practice of the elements. Five chapters deals with "proportion" (analogia), "duplicate, triplicate and compound ratios", and with applications of proportionality to geometry, arithmetic and calculation. The fourth part returs to historiographic questions, and in particulatr, examines reconstructions proposed by earlier authors. The first chapter shows how problematic and uncertain these reconstructions are. The second gives some elements on what became of the euclidean treatment of proportionality in antiquity, the importance of the utilization of the theory of proportions in the "applied mathematics", before and after euclid. The conclusion returns to the status of the elements, and its literary genus, the idiomatic features of which seem to be an (often underestimated) obstacle to the reconstruction of the historical process which led to the elaboration of the concepts and results selected by the author of such a
Abstract FR:
Le travail est organise en quatre parties. La premiere est consacree a l'historiographie; plusieurs problemes importants restent non resolus : nature des objets premiers de la theorie des proportions (rapport ou proportion, nombres entiers et ou grandeurs), histoire de cette theorie avant et apres euclide, mesure de son influence sur les mathematiciens des epoques hellenistiques et romaine. Le premier probleme est repris dans les deuxieme et troisieme parties; la deuxieme examine les objets fondamentaux : nombres et grandeurs et les relations qui peuvent s'etablir entre eux. Deux notions se trouvent mises en avant : celle de multitude (plethos) et celle d'equimultiplicite. A partir de la, le double traitement euclidien de la proportionnalite, considere par la plupart des auteurs comme une difficulte majeure de la structure des elements, est analyse et justifie. La troisieme reprend la question de la thematisation effective des notions de "rapport" et de "proportion" en examinant comment elles interviennent dans la "pratique" des elements. Elle comprend cinq chapitres consacres aux notions de "proportion" (analogia), de "rapports double", "triple" et "compose de rapports", et aux applications de la proportionnalite a la geometrie, a l'arithmetique et au calcul. La quatrieme partie reprend les interrogations historiographiques, en particulier l'examen des reconstructions proposees par la critique anterieure. Le premier chapitre en montre tout le caractere problematique et incertain. Le second chapitre propose quelques elements sur la posterite (dans l'antiquite) du traitement euclidien de la proportionnalite, le role qu'a pu jouer l'ulitisation de cette theorie dans les mathematiques dites appliquees, avant et apres euclide. La conclusion revient sur le statut des elements, sur le genre litteraire auquel le traite appartient, genre dont les caracteristiques paraissent un obstacle (souvent)