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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 Cantor's cardinal numbers represent?

**2.** Where did Euler study at the age of 20?

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

**4.** Which of the following did Dunham concentrate on as one of Newton's great advances?

**5.** Which of the following is a quote from Bertrand Russell included by Dunham?

## Short Essay Questions

**1.** Describe what mathematical and artistic movements are focused on in the second half of the 19th century.

**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.** Describe some of the characteristics of Leonhard Euler, and what made him successful.

**4.** Describe who were Jakob and Johann Bernoulli.

**5.** Describe what the Bernoullis discovered about series, and give an example.

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

**7.** Describe Cantor's difficult personal life.

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

**9.** What was the great theorem of this chapter? Describe it briefly.

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

## Essay Topics

Write an essay for ONE of the following topics:

### Essay Topic 1

Describe how Gerolamo Cardano became involved in the solution to cubic equations. Why was their a controversy over his publication of the solution to cubic equations? Explain.

### Essay Topic 2

In all of the great theorems presented in Dunham's book, which theorem do you think made the most impact on the history of mathematics? Write an essay to defend your opinion with reference to Dunham's descriptions and conclusions.

### Essay Topic 3

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?

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