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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. Which words best describe how solid proofs were developed in Elements?

2. Which mathematician was first to take the challenge to solve cubic equations?

3. What did the Pythagorean Theorem accomplish for mathematics?

4. Who's method did Tartaglia's challenger use in the contest to solve cubic equations?

5. What is a "depressed cubic"?

Short Essay Questions

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

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

3. What was Euclid's definition of composite and perfect numbers?

4. What was already known about circles before Archimedes?

5. Explain what Neils Abel proved.

6. Explain what Archimedes went on to study after the circle, and what was Dunham's opinion of this work.

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

8. Who was Luca Pacioli, and what controversy was sparked over his writings?

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

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 parts of Euclid's Elements. Use the following questions to guide your writing. What were the basic components to Elements? What were the most important theorems and proofs in Elements? How did Elements change the future of mathematics. Name a few mathematicians after Euclid who used Elements in their own work.

Essay Topic 2

Write a three part essay to explain Archimedes determination of circular area.

Part 1) According to Archimedes what was pi, and why is this value needed to determine circular area?

Part 2) Explain how Archimedes used a right triangle to determine the area of a circle.

Part 3) Why was the determination of circular area useful at the time of Archimedes, and how did it advance the future of mathematics?

Essay Topic 3

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.

(see the answer keys)

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