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M4005 Geometry

January 25, 2006

Topics discussed in class

1. Proof in Euclid. The proof of Proposition 2 illustrates the main themes: that everything builds from axioms and defintions. The proof of Proposition 4 shows that Euclid was not perfect.
2. Structure of ancient Greek mathematical exposition. The parts of a proposition and its proof:
• protasis (enunciation)
• ekthesis (setting out)
• diorismos (definition of goal)
• kataskeuē (construction)
• apodeixis (proof)
• superasma (conclusion)
3. Structure of Euclid Book I.

What we did

I lectured most of the period. We began with a review of the idea of definition and of the idea of postulate. Then we studied the statement of Proposition 2. I conducted a dialogue with the class in which the class successfully rediscoved Euclid's argument. This provided an opportunity to highlight some important ideas about formal reasoning--in particular, the need for stract adherence to explicit rules. After this, we studied the structure of the proof of Proposition 2 that Euclid presented, noting the occurrence of the parts in the list above. During the last 40 minutes of class, I went rapidly through all the propositions in Book I, showing how they fell into groups concentrating on a sequence of topics. Propositions 1--26 had several clusters treating basic constructions, basic properties of triangles, lines and angles, facts about the relative sizes of segmants and angles in triangular configurations culminating with the congruence criteria ASA and AAS. With Proposition 27, the Parallel Postulate enters the development. Properties of parallelograms are developed, areas are studied and the Pythagorean Theorem is proved.

What to think about for next class

• Similarity in Stillwell. What Stillwell calls "Thales Theorem" appears in Euclid as Proposition 2 of Book VI. Why does it take until Book VI to present this basic idea? Actually, only the definition of ratio from Book V is needed (for VI Proposition 1). The other ingredient is Proposition 35, which I mentioned as the proposition that Newton found so interesting.
• I will want you to think about the theorems of Pappus and Desargues. Problems in Stillwell that ask you to derive these theorems from Thales will eventually be assigned as homework.

Homework

1. Read the following articles about definitions. Read about definitions here.

2. Problems 1--5 on the January 25 handout (available for download here) will be due next Wednesday. You should start work NOW and come to class on Monday with questions.

3. Learn the proof of Eulid Book I Proposition 35 and Proposition 47.

4. Read (yet again) Stillwell, chapter 1.