On Sunday, 2 March 2014 19:19:08 UTC+2, Jürgen R. wrote: > "John Gabriel" schrieb im Newsbeitrag > > news:email@example.com... > > > > 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. > > > > Bertrand Russell
Russell was an incompetent fool next to Euclid. I think you just helped my argument. :-)
To say that the Elements was a masterpiece of logic has been very grossly exaggerated is to say that Russell was a complete imbecile.