"John Gabriel" schrieb im Newsbeitrag news:firstname.lastname@example.org...
Listed below are the Axioms of magnitude:
Book V, Def. 1: 1. A magnitude is the idea of size of extent. We can either tell that two magnitudes are equal or not. If we can tell they are not equal, then we know which is smaller or bigger, but we can't tell how much bigger or smaller. This is called qualitative measurement (without numbers).
Book V, Def. 3: 2. We can form ratios of magnitudes. AB : CD where AB and CD are line segments. The expression AB : CD means the comparison of magnitudes AB and CD.
Book VII, Def. 1 3. A ratio of equal magnitudes, say AB : AB or CD : CD allows us to use either as the standard of measurement, that is, the unit. The unit is a ratio of equal magnitudes.
Book VII, Def. 2 4. The unit enables us now to compare AB and CD if both are exact multiples of the unit that measures both. We can now perform quantitative measurement, because we can tell how much greater AB is than CD or how much less AB is than CD.
Book VII, Def. 3 - 5 5. Finally, if a magnitude is only part of a unit, then we arrive at a ratio of numbers, say AB : CD where AB and CD are multiples of the unit. AB : CD now means the comparison of numbers AB and CD. When we write AB/CD, it is called a fraction.
So in fact, anyone who disagreed with me, was actually disagreeing with Euclid himself! Tsk, tsk. You should now learn from the terrible folly of your ways. :-)
It has been customary when Euclid, considered as a text-book, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not always define, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid's earlier proofs fail before this test.
...[several examples of the above]
Many more general criticisms might be passed on Euclid's methods, and on his conception of Geometry; but the above definite fallacies seem sufficient to show that the value of his work as a masterpiece of logic has been very grossly exaggerated.