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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.** Heron devised which of the following methods?

**2.** What was the bases of Hippocrates's proof ?

**3.** In Elements, how many postulates must be accepted as given?

**4.** Which of the following was NOT one of the basic definitions in Elements?

**5.** What name did Euclid give for numbers that could be divided by numbers other than themselves and one?

## Short Essay Questions

**1.** What did Euclid state for his theory on prime numbers?

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

**3.** Explain what Neils Abel proved.

**4.** What did Dunham explain about the shift in learning from the West to the East?

**5.** Explain when the knowledge of ancient scholars was rediscovered.

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

**7.** Describe Euclid's definition of prime numbers and the relationship he stated as existing between prime and composite numbers.

**8.** Who was Luca Pacioli, and what controversy was sparked over his writings?

**9.** What was already known about circles before Archimedes?

**10.** Explain some puzzles suggested by Euclid's theories.

## Essay Topics

Write an essay for ONE of the following topics:

### Essay Topic 1

Describe Gauss's work on what was to be known as non-euclidean geometry. What was Gauss's system for triangles where angles added up to fewer than 180 degrees? What were some of his conclusions? Did he publish his work? Was their any controversy surrounding his work on this system? Explain.

### Essay Topic 2

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 3

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.

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