MAT 300-01 (non-L)              Midterm Exam                         

April 26, 2004                          Take-home part

 

This part of the exam is due on Thursday, April 29, at the beginning of class.  These questions are part of the midterm exam. You may use only your text, your notes, and any other resources that have been assigned in class.  You may not receive help from anyone else on these questions, except that you may ask Dr. Gingrich for clarifications.  You also may not give help to anyone else in the class, and you may not talk to anyone else, whether in the class or not, about the test until it has been turned in. 

 

8.         (20 points)        Write an essay on one of the following topics.  Your answer should be  250-400 words typed and double spaced. 

 

a.       A crisis was caused in Greek mathematics by the discovery of the square root of 2 and other irrational numbers.  Explain why this might be called the “commensurability crisis”.  Carefully explain the relationship of this problem to the theory of proportions and explain how the crisis was "resolved".  

b.       Discuss the works of Archimedes.  Your answer should include something about major themes in his work, and the effect and/or influence of his works.  You should name several specific works and briefly discuss their contents.

 

 

9.         (16 points)        Choose two of the following and write your solutions on clean sheets of white paper.  You do not need to type the solution to these problems.

 

 

a.         Give a geometric interpretation or demonstration of the following statement using the techniques of the Pythagoreans.  You should explain in words what the picture represents. Include any auxiliary lines used to explain the reasoning in the proof.
 

  A square number is the sum of two consecutive triangle numbers.

 

b..        For the following proposition from Euclid's Elements, interpret the statement, if appropriate, and indicate its algebraic or modern equivalent.  Include a picture, but you do not need to give a formal proof.

Book II, Proposition 1
If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight lines and each of the segments,

 

c.                   Explain how the method of exhaustion is used by Archimedes in the proof of Proposition 1 in “On the Measurement of the Circle”.

 

 

 

 

THE END!