Re-examination of the different origins of the arithmetical books of Euclid's Elements

Abstract

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.