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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 do we know in modern times about Heron?

**2.** Who was Heron?

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

**4.** What did Dunham consider as Archimedes's "masterpiece"?

**5.** Heron devised which of the following methods?

## Short Essay Questions

**1.** Describe in two sentences Archimedes's method for determining circular area.

**2.** What was Euclid's definition of composite and perfect numbers?

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

**4.** What did Dunham describe in the epilogue of the chapter?

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

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

**7.** Explain what Neils Abel proved.

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

**9.** Explain why Archimedes finding a number value for pi was considered a great achievement according to Dunham.

**10.** 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.

## Essay Topics

Write an essay for ONE of the following topics:

### Essay Topic 1

Write an essay to describe what was known about mathematics and geometry before and in the time of Thales.

Part 1) Describe what we know about geometry before Thales.

Part 2) In what ways were mathematics and geometry used before Thales?

Part 3) Explain what we know about Thales' work. How did it advance mathematics?

### Essay Topic 2

Write a three part essay to compare the work of Euclid to the work of Gauss.

Part 1) Explain what concepts both Euclid and Gauss worked with, and how they approached a similar problem.

Part 2) Compare what Euclid believed about triangles to what Gauss believed about triangles.

Part 3) How did both Gauss and Euclid advance mathematical understanding in their own time?

### 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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