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Name: _________________________ | Period: ___________________ |

This test consists of 15 multiple choice questions and 5 short answer questions.

## Multiple Choice Questions

**1. Dunham showed that Heron's proof could also be used as which of the following?**
**(a)** A proof of Archimedes' number theory. **(b)** A proof of the Pythagorean Theorem. **(c)** A proof of Euclid's number theory **(d)** A proof of Hippocrates' squared areas.

**2. Which of the following was NOT true about Archimedes, according to Dunham?**
**(a)** His death was described as being by the hands of a Roman invader. **(b)** Archimedes died because he refused to follow orders until he completed a math problem. **(c)** He died during the fall of Syracuse. **(d)** He died as a soldier.

**3. In Elements, how many postulates must be accepted as given?**
**(a)** Twenty-two. **(b)** Five. **(c)** Eighteen. **(d)** Twelve,

**4. What shape was NOT demonstrated in the Elements as having a relationship to other shapes?**
**(a)** Pentagon. **(b)** Hexagon. **(c)** Hyperbola. **(d)** Triangle.

**5. How many definitions were stated in Elements?**
**(a)** Five. **(b)** Thirty. **(c)** Twenty-three. **(d)** Eighteen.

**6. Which shapes as described by Euclid, inspired the Greek philosopher Plato?**
**(a)** Regular polyhedrons. **(b)** Manifolds. **(c)** Triangles. **(d)** Spheres and cones.

**7. 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)** If a new number is found to be composite, then it must have some prime as a divisor. **(c)** Divide a infinite group of primes by the sum of their composites. **(d)** After summation, the new number can be prime or composite.

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

**9. Which of Euclid's postulates troubled many of the following generations of mathematicians?**
**(a)** Euclid's postulate on creating an arc. **(b)** Euclid's postulate on right triangles. **(c)** Euclid's proof on right triangles. **(d)** Euclid's postulate on parallel lines.

**10. Which of the following becomes an important definition in mathematics that was first presented in Elements?**
**(a)** 180 degree angle. **(b)** Circle. **(c)** Parallel line. **(d)** Intersection.

**11. What was true about Heron's theorem as described by Dunham?**
**(a)** It was to determine the area of a circle by measuring a right triangle inside the circle. **(b)** It was to find the area of a triangle when only the length of the sides are known. **(c)** It was to determine the volume of a sphere without measuring the circumference. **(d)** It was to solve equations were only two varibles are known.

**12. What provided most of the content in the book Elements?**
**(a)** Hypotheses. **(b)** Postulates. **(c)** Propositions. **(d)** Notions.

**13. How did Archimedes demonstrate his theory of pi?**
**(a)** He demonstrated that the area of the circle is never less than the area of the triangle. **(b)** He demonstrated that the area of the circle is never equal to the area of the triangle. **(c)** He demonstrated that the area of the circle is neither greater than nor less than the area of the triangle and therefore must be equal to it. **(d)** He demonstrated that the area of the circle is always greater than the area of the triangle.

**14. Which of the following was NOT one of Gauss' discoveries?**
**(a)** That there is no apparent contraction to the assumption that the sum of angles in a triangle can have fewer than 180 degrees. **(b)** That angles in a triangles can not add up to more than 180 degrees. **(c)** "Non-euclidean" geometry. **(d)** That under Euclid's definition parallel lines can intersect.

**15. Which of the following is an example of a postulate that must be accepted in Elements?**
**(a)** It is possible to draw a straight line between an infinite number of points. **(b)** It is possible to connect any two points with a line and make a circle. **(c)** It is possible to draw an arc with any three points. **(d)** It is possible to draw a circle that contains no lines.

## Short Answer Questions

**1.** What name did Euclid give for numbers that could be divided by numbers other than themselves and one?

**2.** What did Dunham consider as Archimedes's "masterpiece"?

**3.** Who was Heron?

**4.** In what century did Archimedes live?

**5.** What was true about Hippocrates's proof?

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