1. What remarkable facts did Archimedes discover about the cylinder vs the sphere?

2. How did Archimedes discover and how did he prove various geometric facts? Describe his methods, and outline examples of their application.

3. What is the Theorem of the Broken Chord? What fact about angles in a circle is used to prove it?

4. What is Ptolemy’s Theorem? Prove it.

5. State a trig identity related to Ptolemy’s Theorem, and show how it is derived.

6. Describe a famous problem in the Arithmetica of Diophantus, with its more recent history.

7. Describe the stages in the development of algebra outlined in our text, with some names, dates, and places.

8. Outline the development of some of the algebraic notation we use today, with names and dates.

9. What was the principal importance of Arab mathematics during the middle ages?

10. Solve the following problem using the rule of false position. A quantity, its half, its third, and its twelfth are added together to give 184. What is the quantity?

11. Outline some important events in the history of computation, from its beginnings to the times we have finished studying at this point in the course. Names, approximate dates, events, geography will earn points.

12. Do the following computations, assuming that all computations are being done BASE 6.

a) 34 + 12 b) 452 + 5324 c) 22 × 153

13. Outline the history of the solution of polynomial equations from ancient times to the present point of the course. Events, names, dates, places, events will earn points.

14. Name two or more Arab mathematicians (spelling is not vital), and describe when and why they are important.

15. Describe the Fibonacci sequence, and some mathematics connected with it.

16. What important computing tools have we studied? Describe them, approximate dates, names when possible, and how they were important.

17. Describe some mathematical advances in astronomy studied in the course, naming names and giving approximate dates.

18. Explain Cavalieri congruence, and apply it to an example.

19. Select two people from the following list of three and describe their geography, timeframe, and mathematical contributions. [Names could include Archimedes, Ptolemy, Diophantus, Edscartes, Euler, Omar Khayyam, al-Khowarizmi, Fibonacci, Cardano, Tartaglia, Briggs, Napier, Galileo, Kepler, Cavalieri].

20. Use the method of similitude to find the largest 2:3:4 triangle in a given semicircle.