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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 series was Euler most famous for?

**2.** What did most of 19th century mathematics focus on, as highlighted by Dunham?

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

**4.** What did mathematicians want to perfect in the mid-19th century?

**5.** Who eventually solved the sum of the successive squared denominator series?

## Short Essay Questions

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

**2.** Give an example of a series who's sum is still unknown.

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

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

**5.** Explain why Eulers sum of Ï€Â²/6 was in some ways surprising.

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

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

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

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

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

## Essay Topics

Write an essay for ONE of the following topics:

### Essay Topic 1

Write a three part essay to compare Archimedes's and Newton's determination of a number value for pi.

Part 1) Explain what general approach each mathematician used in the determination of pi.

Part 2) What prior knowledge of mathematics is required to use each method?

Part 3) Which number determination for pi was more accurate, Newton's or Archimedes? Explain.

### Essay Topic 2

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 3

Debate in essay format one of the following controversies:

a) Newton versus Leibniz.

b) Cardano versus Tartaglia.

Explain a defense for each side of the controversy, then propose your opinion with your final conclusions.

This section contains 836 words(approx. 3 pages at 300 words per page) |