Pons asinorum: what's wrong with this proof?
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From: bert on 7 Aug 2010 10:24 On 7 Aug, 15:19, Chip Eastham <hardm...(a)gmail.com> wrote: > A good account of Euclid's proof and analysis > of why there are no "shortcuts" to the result > is here: > > http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI5.html Well, Pappus's proof looks exactly the same as that anecdotal automatically-generated proof, and it looks like a pretty "shortcut" to me. --
From: kj on 8 Aug 2010 07:22 In <1ffd4e45-31d8-4af8-80ed-b882edf9f1c8(a)f6g2000yqa.googlegroups.com> Chip Eastham <hardmath(a)gmail.com> writes: >In proving the "interior" base angles of the >isoceles triangle are equal, Euclid can and >does make use of SAS congruence (of the >triangle to itself, in essence) just proven >in Prop. 4 of Book I. However Prop. 5 has >a second conclusion, that "exterior" base >angles, the result of extending the equal >sides beyond the base, are also equal. It >"proves" to be a good bit more subtle to >establish. The subtlety is completely lost on me! It seems obvious that if any two angles are equal (and, in particular, if the two base angles are equal) their complements (relative to the straight line) are also equal. IOW, if a + a' = b + b' = Pi radians, and we have already proven that a = b, then it follows that a' = b'. Again, it may be that this "obvious" theorem is just out of place in Euclid's sequence, but if so, his sequence was not very well thought out...
From: I.M. Soloveichik on 8 Aug 2010 05:24 > http://aleph0.clarku.edu/~djoyce/java/elements/bookI/p > ropI5.html According to that discussion, Euclid had not yet shown that all straight angles are equal so he couldn't conclude exterior angles of equal angles are equal.
From: kj on 8 Aug 2010 11:01 In <525265751.75809.1281273874692.JavaMail.root(a)gallium.mathforum.org> "I.M. Soloveichik" <imsolo47(a)yahoo.com> writes: >> http://aleph0.clarku.edu/~djoyce/java/elements/bookI/p >> ropI5.html >According to that discussion, Euclid had not yet shown that all >straight angles are equal so he couldn't conclude exterior angles >of equal angles are equal. I had not read that claim, but now that you point it out, it seems to me very lame. To me the equality of straight angles verges on "definitional", just like the "equality" of right angles, and the equality of angles measuring 12345/67890 radians, etc. At any rate, if their claim is true, then the ordering Euclid chose for his axioms and postulates is silly at best: the equality of straight angles is surely more fundamental and useful than the "pons asinorum" postulate.
From: Chip Eastham on 8 Aug 2010 17:22
On Aug 8, 11:01 am, kj <no.em...(a)please.post> wrote: > In <525265751.75809.1281273874692.JavaMail.r...(a)gallium.mathforum.org> "I..M. Soloveichik" <imsol...(a)yahoo.com> writes: [Euclid's Elements: Book I, Prop. 5 -- D. E. Joyce] http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI5.html > >According to that discussion, Euclid had not yet shown that all > >straight angles are equal so he couldn't conclude exterior angles > >of equal angles are equal. > > I had not read that claim, but now that you point it out, it seems > to me very lame. To me the equality of straight angles verges on > "definitional", just like the "equality" of right angles, and the > equality of angles measuring 12345/67890 radians, etc. As the link offered several times explains, Euclid does not use the notion of "straight angle", though as you seem to be aware, he does define "right angle" and then postulate the equality of all right angles. However this definition is not a matter of "measure" of angles in radians, as may seem natural to your modern view. See in Book I the Definition 10 and Postulate 4 (all right angles are equal). Joyce points out that to deduce the equality of the "exterior" base angles in Prop. I.5 from the equality of the (supplementary) "interior" base angles by an appeal to Prop. I.13 would be circular. The proof of Prop. I.13 depends on Prop. I.7, and its proof in turn depends on Prop. I.5. > At any rate, if their claim is true, then the ordering Euclid chose > for his axioms and postulates is silly at best: the equality of > straight angles is surely more fundamental and useful than the > "pons asinorum" postulate. I hope you mean "pons asinorum" proposition, since Euclid accomplishes its proof rather than merely its assumption. There is much value to reducing the number of primitive notions and postulates (assumptions) to a minimum, so that the extent of deductive reasoning is laid bare. While there are points in Euclid's development in which gaps are evident (cases whose consideration is omitted, reliance on unstated postulates or notions, and in a few places letting the picture dictate a conclusion), I think Euclid chose well in making the right angle a central notion rather than a straight angle. It can be tempting for us who were brought up on the 17th century Cartesian notions of coordinate geometry to criticize the historical foundation laid by Euclid and his predecessors. But I think history has shown the Elements to be a model of rigor through ages in which little new was being added or even sought in the way of mathematics. regards, chip |