Honors Interdisciplinary Seminar (175)

 Euclid's Elements

Assignment following Week #7 (Due 11/2 in class)
(Section 004 - Due 9/21 in class) 

  1. Carefully read background material on Euclid found in the short excerpt from Greenberg’s text Euclidean and non-Euclidean Geometry:  (ereserve)
  2. Carefully read the first book of Euclid’s Elements, focusing on Propositions 1- 20, 47, and 48.  Make sure you carefully read the proofs as well as the statements.  Two sources:

    a) Densmore's edition of the Heath translation (ereserve:  See "Euclid's elements, book I")

    b) David Joyce’s interactive Elements at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html 

 I recommend printing out Densmore’s version of Book I and bringing it to our next class. Joyce's web page provides nice links and supplementary materials and is easy to read online.

  1. Complete the study questions listed below.
  2. With your partner(s), prepare your proof of the assigned proposition that you will present at our next class meeting.  You can find parameters, recommendations, and a sample evaluation form for your presentation HERE.  Remember to email me (szydliks@uwosh.edu) or see me if you have questions.
  3. Read any emails I send you, and respond if requested.
  4. Bring a straightedge and compass to our next class.  There may be a brief quiz at the start of class.  If there is a quiz, it will be based on your reading, as a simple check that you have read the material reasonably carefully.  Here is a sample reading quiz.  I recommend that you look at it only *after* you have completed the readings, as a check of your readiness.

Study Questions 

Please carefully write out your answers to these questions.  Make a copy of your answers and be prepared to hand that copy in at the start of class.  Look at the Study Questions Main Page for general guidelines on study questions in Steve's section.

Followup on “People and Clubs”: 

  1. Suppose there are n people living in Hilbert. Is it ever possible that there is a club which contains n-1 people?  Either give an example to show that it is possible, or give a careful and complete argument why it is always impossible.

    (If you find this "general argument" too hard, then do the following problem instead:  Suppose that there are 8 people living in Hilbert.  Is it ever possible that there is a club that contains 7 people?  Either give an example to show that it is possible, or give a careful and complete argument why it is always impossible.)
  2. Find another possible population for Hilbert, other than those possibilities we discussed in class (0, 1, 2, 3, 4). Show that your population really is possible, by giving a valid “club system” for that number of people.

From the Euclid readings:

  1. Why are Euclid’s Elements considered such a monumental achievement?  Be specific.
  2. Criticize Euclid’s definitions.  Are there any terms that you think he did an especially good job or an especially poor job of defining?  Make a note of those.
  3. How do both Heath/Densmore and Joyce use the word "line" to mean something different from our usual terminology?   What is a "rectilinear (or rectalineal) angle"?
  4. Write down each of Euclid’s 5 postulates.  Draw a picture of each postulate and give an explanation in your own words.
  5. In your own words, what does Proposition 4 (IV) say?
  6. Propositions 1,2,3, 9, 10, 11, 12,  and several others are different from the other Propositions.  (For example, Joyce uses different colors for them on his web page.)  Why?  What makes them different?
  7. Carefully read the statement of Proposition 47 (XLVII).  What does it say in more familiar terms?  How are the statement and proof different from the way you usually see the proposition?
  8. What does the "People and Clubs" activity have to do with Euclid?  Explain.

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