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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.** Which of the following is false about the modern implications of Euclid's number theory?

**2.** According to Dunham, who was most able to collect knowledge from around the globe?

**3.** Which words best describe how solid proofs were developed in Elements?

**4.** In general, what did Euclid's number theory describe?

**5.** Which of the following is an example of a perfect number?

## Short Essay Questions

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

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

**3.** Describe the events that follow del Ferro's death between Fior and Tartaglia.

**4.** Explain who was Hippocrates, his contribution to mathematics, and how do we know about him?

**5.** Explain what Archimedes went on to study after the circle, and what was Dunham's opinion of this work.

**6.** Describe how Cardano eventually publishes the solution to cubic equations.

**7.** Explain in two sentences Euclid's method to prove the Pythagorean Theorem.

**8.** Describe the contents of Cardano's book.

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

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

## 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

Write a three part essay to explain Archimedes determination of circular area.

Part 1) According to Archimedes what was pi, and why is this value needed to determine circular area?

Part 2) Explain how Archimedes used a right triangle to determine the area of a circle.

Part 3) Why was the determination of circular area useful at the time of Archimedes, and how did it advance the future of mathematics?

### Essay Topic 3

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.

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