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This test consists of 5 short answer questions, 10 short essay questions, and 1 (of 3) essay topics.

Short Answer Questions

1. What were the proofs in Elements based on?

2. What did Gauss set out to prove?

3. Numbers whose divisor add up to itself, was considered which type of number according to Euclid?

4. Which was true of Euclid's number theory?

5. After Hippocrates, what shape did the Greeks attempt to square without success?

Short Essay Questions

1. Describe Lindeman's work on the square of a circle, and state what he discovered.

2. What was already known about circles before Archimedes?

3. How did Heron find the area of a triangle, and what did Dunham state about Heron's work?

4. Describe why Dunham infers that Euclid's Elements was an evolutionary book not so much for what it said, but in how it was presented.

5. Describe who was Archimedes and how Dunham described his character.

6. Why did Euclid's postulate on parallel lines trouble mathematicians for centuries?

7. Describe Euclid's postulates and notions in how they were important in constructing his proofs.

8. Describe what is quadrature and why it was useful in the time of Hippocrates.

9. How many definitions were in Euclid's book? List some of the definitions he included.

10. Who was Heron, and what is known about him today?

Essay Topics

Write an essay for ONE of the following topics:

Essay Topic 1

Compare Pythagoras's proof of the Pythagorean Theorem and how Heron's formula for triangular area can be used as an alternative proof of the Pythagorean theorem. What are the common assumptions in both proofs? What components of each proof are different?

Essay Topic 2

Describe how Gerolamo Cardano became involved in the solution to cubic equations. Why was their a controversy over his publication of the solution to cubic equations? Explain.

Essay Topic 3

Debate in essay format one of the following controversies:

a) Newton versus Leibniz.

b) Cardano versus Tartaglia.

Explain a defense for each side of the controversy, then propose your opinion with your final conclusions.

(see the answer keys)

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