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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 did Euler's sum surprisingly connect?

**2.** In the Bernoulli's time, what was the current definition of a series?

**3.** What did Dunham describe as the same between artistic movements and mathematical studies in the 19th century?

**4.** When was Euler born?

**5.** Where does the center of mathematical thinking shift to after Italy?

## Short Essay Questions

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

**2.** Describe the connection between Fermat and Euler's work.

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

**4.** What was Gauss's major unpublished achievement in geometry?

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

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

**7.** Describe the great theorem explained by Dunham in this chapter.

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

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

**10.** Explain what was the definition of a series before the Bernoullis, and give examples of what was known.

## Essay Topics

Write an essay for ONE of the following topics:

### Essay Topic 1

Explain the accuracy of Archimedes determination of pi. How did he determine the number value of pi? How accurate is Archimedes's measurement of pi? What other mathematician was able to determine a number value, though more accurate, for pi?

### Essay Topic 2

Summarize the discoveries on series made by the Bernoulli brothers and later, Euler. How did the Bernoullis advance mathematical understanding of an infinite series, and how did Euler even further advance this knowledge? Give examples. What principles about series and the sum of series are still being worked out in modern 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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