Journey Through Genius: The Great Theorems of Mathematics Test | Mid-Book Test - Hard

William Dunham (mathematician)
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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 Euclid do in his 48th proposition?

2. What did most of Heron's work deal with?

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

4. Which of the following was NOT one of Gauss' discoveries?

5. Who was the author of the book Elements?

Short Essay Questions

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

2. How did Archimedes find a number value for pi?

3. Describe the great theorem explained by Dunham in this chapter.

4. Describe how Cardano eventually publishes the solution to cubic equations.

5. Explain why Eulers sum of π²/6 was in some ways surprising.

6. What did Dunham explain about the shift in learning from the West to the East?

7. Why did Euclid's postulate on parallel lines trouble mathematicians for centuries?

8. Describe who were Jakob and Johann Bernoulli.

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

10. Describe what work of Euclid's fascinated Plato and his theory on the shape of the Universe.

Essay Topics

Essay Topic 1

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 2

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 3

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.

(see the answer keys)

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