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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.** Where was Euler born?

**2.** When was Euler born?

**3.** Who else, besides Newton, independently discovered a calculus method?

**4.** Which word best describes Newton's childhood?

**5.** Which of the following was a major part of Gauss' work in mathematics?

## Short Essay Questions

**1.** Describe the controversy that Newton was caught in with his publication of his calculus methods.

**2.** Where did the center of mathematical thinking shift to in the !7th Century, and who are the major scholars of this time period?

**3.** Who was Georg Cantor, and what was significant about his work in mathematics?

**4.** Describe some of Gauss's work.

**5.** Why did Euler start working on the sum of series?

**6.** Explain how Gottfried Leibniz was able to publish his method of calculus.

**7.** Describe Newton's days in Cambridge and what he eventually came to discover.

**8.** Explain any methods used by Cantor that were unsuccessful.

**9.** Summarize in a few sentences, what types of number sets did Cantor prove to be denumerable and non-denumerable.

**10.** What great theorems and work of Newton did Dunham highlight?

## 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 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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