By Cochran T.
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Extra resources for 4-Manifolds which embed in R5 R6, and Seifert manifolds for fibered knots
To begin with, he constructs arithmetical ways of generating these sequences. He then strives to establish relationships between these sequences and the first sequences of polygonal numbers (triangular, square, pentagonal and so on).
Netz’s analysis distinguishes three types of intervention that, in his words ‘produce[d] an Archimedes who was textually explicit, consistent, rigorous and yet opaque’. In particular, Netz’s overall broader argument reveals how Heiberg shaped Archimedes’ 37 Saito 2006: 97–144 compares Heiberg’s diagrams in Book i of the Elements with those of the Greek manuscripts which formed the basis of his critical edition. Mathematical proof: a research programme proofs according to his vision. In conclusion, we understand better how we were mistaken, when we took Heiberg’s words for Archimedes’ writings as the manuscripts bear witness to them.
The second systematic intervention by Heiberg which Netz analyses is the bracketing of words, sentences and passages in Archimedes’ writings, despite the fact that the manuscripts all agree on the wording of these passages. In other words, by rejecting portions as belonging to the original text, Heiberg modified the received text of Archimedes’ writings in conformity with the representation that he had formed for Archimedes as a sharp contrast to Euclid. While, for Heiberg, Euclid was characterized by the careful expression of the full-fledged argument, Archimedes’ style was, in his view, to focus on the main line of the proof, leaving aside ‘obvious’ details.