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. Exactly what limit is reached at a quartic equation?

2. Which shapes as described by Euclid, inspired the Greek philosopher Plato?

3. What did Hippocrates do that advanced mathematical methods?

4. What else, besides a solution to cubic equations, was in Cardano's book?

5. What do we know in modern times about Heron?

Short Essay Questions

1. Describe Lindeman's work on the square of a circle, and state what he discovered.

2. Describe Euclid's definition of prime numbers and the relationship he stated as existing between prime and composite numbers.

3. Explain some puzzles suggested by Euclid's theories.

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

5. Describe why Dunham infers that Euclid's Elements was an evolutionary book not so much for what it said, but in how it was presented.

6. Describe in two sentences Archimedes's method for determining circular area.

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

8. What was already known about circles before Archimedes?

9. Describe the contents of Cardano's book.

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

Essay Topics

Write an essay for ONE of the following 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

Summarize the puzzles that were presented as a result of Euclid's infinitude of primes. Name and describe some of these puzzles from a historical perspective, then explain what is known about each in modern mathematics. What puzzles remain unsolved today?

Essay Topic 3

Write an essay to describe what was known about mathematics and geometry before and in the time of Thales.

Part 1) Describe what we know about geometry before Thales.

Part 2) In what ways were mathematics and geometry used before Thales?

Part 3) Explain what we know about Thales' work. How did it advance mathematics?

(see the answer keys)

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