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Article ID: CBB411050920

Re-Examination of the Different Origins of the Arithmetical Books of Euclid's Elements (2019)

Saito, Ken (Author)

Historia Mathematica
Volume: 47
Pages: 39-53

DOI: 10.1016/j.hm.2019.03.002

URI: https://www.sciencedirect.com/science/article/pii/S0315086018301435

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A quick examination of the diagrams in the Greek manuscripts of Euclid's Elements shows that VII.28 is the only proposition which has a diagram with both horizontal and vertical lines, and it is quite similar to parallel propositions in Book X (X.15, 16). This leads to an audacious assumption that all the propositions of Book VII after it may have been added later, and their authenticity is examined by logical and linguistic analysis. In many of the propositions of Book VII after VII.28, we have found characteristics or idiosyncrasies which can be explained by the hypothesis of later addition. Moreover, some propositions can be interpreted as lemmata for conspicuous results in Book IX such as IX.20 (the number of prime numbers surpasses any given number), IX.36 (sufficient condition of a perfect number). While the prevalent view is that Book VII is a well-organized basic theory from which various arguments in the subsequent two books are developed, our examination suggests that Book VII itself contains heterogeneous components which are intended for specific arguments in subsequent books. Further investigation is needed to find out whether this heterogeneity in Book VII is the result of later intervention, or that of original compilation. Résumé Un rapide examen des diagrammes dans les manuscrits grecs des Eléments d'Euclide montre que la proposition VII.28 est la seule dans les trois livres arithmétiques (livres VII–IX) qui a un diagramme avec des lignes horizontales et verticales, ce qui est similaire à ceux des propositions parallèles du livre X (X.15, 16). Cela nous a conduit à l'hypothèse audacieuse que toutes les propositions du livre VII après VII.28 peuvent avoir été ajoutées postérieurement. Leur authenticité est examinée au moyen d'analyses logiques et linguistiques. Dans de nombreuses propositions du livre VII après VII.28, nous avons trouvé des caractéristiques ou des idiosyncrasies qui peuvent être expliquées par l'hypothèse d'un ajout postérieur. De plus, certaines propositions peuvent être interprétées comme des lemmes pour des résultats remarquables du livre IX, tels que IX.20 (les nombres premiers sont plus nombreux que tout nombre donné) et IX.36 (condition suffisante pour qu'un nombre soit parfait). Alors que l'opinion dominante est que le livre VII présente une théorie de base bien organisée, à partir de laquelle divers arguments des deux livres suivants se développent, notre examen suggère que le livre VII lui-même contient des composants hétérogènes destinés à des sujets spécifiques dans les livres VIII et IX. D'autres recherches sont nécessaires pour comprendre si cette hétérogénéité du livre VII est ou non le résultat d'interventions postérieures.

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