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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 provided most of the content in the book Elements?

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

**3.** Who asked Tartaglia for his solution to cubic equations?

**4.** Which city was the center of thinking and learning in Third century BC?

**5.** Who was the first of ancient philosophers to consider why geometric properties existed?

## Short Essay Questions

**1.** Explain who was Gerolamo Cardano, and how did he become involved with the solution to the cubic.

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

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

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

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

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

**7.** How did Archimedes find a number value for pi?

**8.** 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.

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

**10.** Explain what Neils Abel proved.

## Essay Topics

Write an essay for ONE of the following topics:

### Essay Topic 1

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 2

Write a a three part essay to explain how Cantor's work in mathematics also fit into the cultural movements of the time.

Part 1) Describe the artistic movements of the 1860s and 1870s. In general, how could they be related to mathematical philosophy at the time?

Part 2) What was Cantor's work in mathematics in the 1860s and 1870s?

Part 3) Explain how Cantor's philosophy and mathematical theorems fit with the new artistic and philosophic ideas of the 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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