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. How many definitions were stated in Elements?

2. What did Gauss set out to prove?

3. What did Heron's advances put into historical perspective for Dunham?

4. What did Ferdinand Lindeman prove in 1882?

5. Which of Euclid's postulates troubled many of the following generations of mathematicians?

Short Essay Questions

1. Describe Euclid's postulates and notions in how they were important in constructing his proofs.

2. Who was Heron, and what is known about him today?

3. What was already known about circles before Archimedes?

4. Explain in two sentences Euclid's method to prove the Pythagorean Theorem.

5. Describe what is quadrature and why it was useful in the time of Hippocrates.

6. Explain what Neils Abel proved.

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

8. According the Dunham, how did Euclid prove his theory on the infinitude of primes?

9. Describe the ancient city Alexandria and name a few of its third century geniuses.

10. How did Heron find the area of a triangle, and what did Dunham state about Heron's work?

Essay Topics

Write an essay for ONE of the following topics:

Essay Topic 1

Write a three part essay to describe third century Alexandra.

Part 1) Describe the geographic location of Alexandria. Why did its location affect its scholars?

Part 2) Name and describe the work of a few scholars in third century Alexandria.

Part 3) Who was Heron? How did his achievements improve mathematics?

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

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?

(see the answer keys)

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