N.B. If you feel a question is ambiguous or you want to justify the answer you give, feel free to include a note of explanation.
Part I. Circle "T" for True, "F" for False. (10 pts)
T F 1. The Babylonians most likely knew the divide and average technique for computing square roots.
T F 2. A proposition equivalent to the law of cosines is included in Euclid's Elements.
T F 3. Using ruler and compass it is possible to construct a regular polygon with 72 sides.
T F 4. Plato encouraged the study of mathematics because he felt it was good training for future philosophers.
T F 5. The Babylonians had a fully developed place value system very much like the one we use today.
T F 6. Euclid made a distinction between common notions and postulates, but we don't today.
T F 7. The Greeks did not know how to solve quadratic equations.
T F 8. Building pyramids probably didn't require a great deal of mathematics.
T F 9. We know precisely how the Egyptians constructed the 2/n table which is in the Rhind papyrus.
T T 10. The whole plan of Euclid's Elements leads up to Book XIII which deals with the regular solids.
Part II. Match the items in the second column with those in the first in the best possible way.
____ Archimedes 1. Regular polygons. (20 pts)
____ Babylonians 2. The first woman mathematician.
____ Dinostratus 3. Believed that all is number.
____ Gauss 4. Discovered the Pythagorean theorem.
____ Hippasus 5. Trisected angles using the first curve.
____ Hippias 6. Eudemian summary.
____ Hypatia 7. His geography influenced Columbus.
____ Proclus 8. Wrote the Sand Reckoner.
____ Ptolemy 9. His brother was also a mathematician.
____ Pythagoreans 10. Duplication of the cube
____ Ahmes 1. Espoused atomism.
____ Archytas 2. Use straightedge and compass only.
____ Babylonians 3. Method of exhaustion.
____ Democritus 4. The Rhind Papyrus.
____ Euclid 5. Established the quadrivium.
____ Eudoxus 6. Wrote a lost book on Optics.
____ Hippocrates 7. Plympton 322.
____ Merzbach 8. An author of our book.
____ Plato 9. Had a theory of Proportion.
____ Pythagoreans 10. Quadrature of lunes.
Part III. Arrange the following names in chronological order, from the earliest (1) to the most recent(7). (10 pts)
____ Archimedes
____ Boyer
____ Euclid
____ Eudemus
____ Eudoxus
____ Hilbert
____ Neugebauer
____ Proclus
____ Thales
____ Wantzel
Part IV. Fill in the blanks. (15 pts)
1. One of the greatest mathematicians of all time and certainly the greatest of antiquity was ________________ .
2. Our knowledge of Babylonian mathematics dates from about the year A.D.______.
3. Euclid's Elements contains ______ books and was written about the year ____________ AD or BC (circle one).
4. Euclid's Fifth Postulate is also known as the _______________.
5. The Rhind Mathematical Papyrus was written about _____ B.C.
6. One of the discoverers of Non-Euclidean Geometry was ________________ .
Part IV. Put an "x" in the blank before EVERY alternative for which the statement is true:
1. Developments in ancient mathematics have been inspired by: (15 pts)
____ A desire to explain physical phenomena.
____ Economic stimuli.
____ Intellectual curiosity.
____ Unsolved problems. ____ Previous mathematical developments. 2. Euclid's Elements included propositions on: ____ Number theory.
____ Solid geometry.
____ The conics.
____ Theory of Proportion.
____ Trigonometry.
____ Incommensurables.
3. The "method of false position" was used by the Egyptians or Babylonians to solve problems for which we now use:
____ Linear equations in one unknown.
____ Quadratic equations.
____ Laws of exponents.
____ Definite integrals.
____ Systems of equations.
Part V. Short Answer questions: (5 pts each)
1 What is the golden section?
2. State one of Zeno's paradoxes.
3. Explain the duplication problem to a non-mathematician.
4. What motivated the invention (or discovery) of the conics? Extra credit for giving the main dissenting view.
5. What is a "double mean proportion" and why is it important?
Part VI. Essay question. Answer one of the following questions, indicating which you choose to answer. Both the content and style of your essay will be evaluated. Use the back of the sheets if necessary. (15 pts)
a) Describe and contrast the evidence on which our knowledge of Egyptian and Babylonian mathematics is based. Do you think it likely that our knowledge will be significantly altered by the discovery of new documents? Explain.
b) Morris Kline has said that Egyptian and Babylonian mathematics had "Some vigor, if not rigor, and more perseverance than brilliance."
Discuss. Part VII. Bonus Questions. (5 points each)
1. We discussed several solutions to the trisection problem. Explain how one of them works, indicating who did the work. Also, explain what is meant by the word solution.
2. Theon of Smyrna, a Neoplatonist and Neo-Pythagorean of the second century, is said to have found that the sum of two consecutive triangular numbers is a square. Prove this theorem.