Journey Through Genius: The Great Theorems of Mathematics Test | Mid-Book Test - Easy

William Dunham (mathematician)
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This test consists of 15 multiple choice questions and 5 short answer questions.

Multiple Choice Questions

1. 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 solve equations were only two varibles are known.
(d) It was to determine the volume of a sphere without measuring the circumference.

2. What was known about pi, during Archimedes' time?
(a) Nothing, the concept of pi was unknown.
(b) That it could not be assigned a relationship between measurements in a circle.
(c) That is was the relationship between the diameter and circumference of a circle.
(d) That it was never the same number value for a given circle.

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

4. How did Archimedes arrive at a number value for pi?
(a) By proving that pi could not be a negative number.
(b) By constructing multi-sided polygons inside and outside a circle and determining their perimeters.
(c) By proving pi could not be equal to one.
(d) By constructing successively smaller circles inside circles until he realized all of their ratios of diameter to area were equal.

5. Which of the following was NOT one of the things Dunham claimed was ingenious about Euclid's proof of the Pythagorean theorem?
(a) Euclid used his own axioms and propositions to show relationships,
(b) Euclid used propositions about similar angles and parallel lines.
(c) Euclid stated that the diagonal hypotenuse of a right triangle is equal to the sums of the squares of the two legs.
(d) Euclid constructed squares on the sides of right triangles.

6. That properties of specific shapes were early Egyptians aware of?
(a) Parallelograms.
(b) Right triangles.
(c) Irregular solids.
(d) Pi and the diameter of a circle.

7. Which was true of Euclid's number theory?
(a) It was incorrect, as proved by Plato.
(b) It has been proven too basic to be useful.
(c) It has an impact on modern math.
(d) It was proven to the true by Hippocrates.

8. Who's method did Tartaglia's challenger use in the contest to solve cubic equations?
(a) Pacioli's method.
(b) del Ferro's method.
(c) Cardano's method.
(d) Fontana's method.

9. What did Archimedes manage to prove using Euclid's ideas?
(a) That the area of a circle and the square of its diameter is really the same as the relationship of diameter to circumference.
(b) That the square of a diameter is equal to pi.
(c) That the relationship of area to circumference is really the same as the relationship of radius to diameter.
(d) That the value of pi is proportional to the area of the circle.

10. What did Apollonius work with in mathematics?
(a) He worked on advanced algebra.
(b) He develped a way to measure the volume of a sphere.
(c) He worked on conics.
(d) He discovered the algrebraic equation.

11. Heron devised which of the following methods?
(a) A way to determine the area of a triangle.
(b) A way to determine the volume in a sphere.
(c) A way to solve binomial equations.
(d) A way to apply mathematics to public health.

12. What were the proofs in Elements based on?
(a) Novel notions.
(b) Ancient greek geometry.
(c) Lindeman's method.
(d) Basic definitions.

13. What was Eratosthanes most famous for?
(a) He showed that the Earth must be a sphere.
(b) He developed a way to navigate using logitude and latitude.
(c) He showed that there are no even prime numbers.
(d) He developed a simple way to find prime numbers and for determining the circumference of the Earth.

14. Which is one of the common notions presented in Elements?
(a) "Things which are equal to the same thing are also equal to each other."
(b) "Points with equal values can be connected with a line of equal value."
(c) "Things with are equal have an inverse that is equal."
(d) "The inverse of a line makes a circle."

15. Who was Eratosthanes?
(a) He was the first to study political sciences.
(b) He was a mathematician, and leading doctor.
(c) He was the chief librarian, and a mathematician.
(d) He was a teacher and philosopher.

Short Answer Questions

1. What was Hippocrates's great advance to mathematics?

2. Which of the following was NOT true about Archimedes, according to Dunham?

3. After working on pi, what did Archimedes continue with in his study of mathematics?

4. What was the title of Cardano's book which contained the solution to the cubic?

5. Which of the following is INCORRECT, and not used in Archimedes proof of his theory?

(see the answer keys)

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