On Euclid’s 'Elements' Book V, Definition 7 - Błaszczyk, Piotr (Philosophical Problems in Science (Zagadnienia Filozoficzne w Nauce), 2010)
Euclid's 'Elements' Book V develops theory of proportion of 'geometric magnitudes'. Definition V,5 is the definition of proportion, A:B::C:D, definition V,7 is the definition of the order of ratios, A:B - C:D. In commentaries on Book V it is usually supposed, and sometimes even proved, that the order of ratios is a total order, while it is also supposed that 'magnitudes of the same kind' obey the Archimedean axiom only, i.e. Euclid's definition V,4. The purpose of this paper is to show that the linearity of the order of ratios cannot be deduced from the Archimedean axiom; to this end we define a structure of magnitudes that obeys the Archimedean axiom and show that the conjunction of negations is satisfied (for mathematical symbols applied see the original paper).