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This test consists of 5 multiple choice questions, 5 short answer questions, and 10 short essay questions.

## Multiple Choice Questions

**1. Which of the following could NOT be included as a step in Euclid's great theorem?**
**(a)** Take a finite group of primes and add them together, plus one. **(b)** After summation, the new number can be prime or composite. **(c)** If a new number is found to be composite, then it must have some prime as a divisor. **(d)** Divide a infinite group of primes by the sum of their composites.

**2. What did Euclid state about pi in Elements?**
**(a)** There is no relationship between the area of a circle and its circumference. **(b)** The proportion of area to circumference is never equal. **(c)** There is a constant relationship between the area of a circle and the square of its diameter. **(d)** The proportion of diameter to area is never equal.

**3. What did Dunham consider as Archimedes's "masterpiece"?**
**(a)** Archimedes' work on determining a number value for pi. **(b)** Archimedes' work on volume to surface area ratios. **(c)** Archimedes' work on shperes, cones, and cylinders. **(d)** Archimedes' work on determining angular measurements.

**4. How did Lindeman prove his conclusion?**
**(a)** Lindeman proved that square roots are irrational numbers. **(b)** Lindeman proved that some numbers are constructable without the use of a compass. **(c)** Lindeman proved that all numbers are constructable with a compass and ruler. **(d)** Lindeman proved that some numbers are not constructable with only a compass and straight-edge.

**5. Which of the following is true in modern math about twin primes?**
**(a)** They are not considered whole numbers. **(b)** We don't know if they are finite or infinite. **(c)** Their sum is always another prime number. **(d)** They are infinite.

## Short Answer Questions

**1.** How many sides did the pentadecagon have, as presented by Euclid?

**2.** Who was Eratosthanes?

**3.** What did Apollonius work with in mathematics?

**4.** In what time period did mathematicians find a solution to cubic equations?

**5.** What shape was NOT demonstrated in the Elements as having a relationship to other shapes?

## Short Essay Questions

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

**2.** Explain when the knowledge of ancient scholars was rediscovered.

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

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

**5.** Explain what Neils Abel proved.

**6.** Explain who was Gerolamo Cardano, and how did he become involved with the solution to the cubic.

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

**8.** Describe the contents of Cardano's book.

**9.** 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.

**10.** What did Euclid state for his theory on prime numbers?

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