MATH 629 -- History of Mathematics
MIDTERM EXAM
March 3, 1997

1. Give three significant aspects on the development of ancient mathematics that resulted from the limitations of printed material.
2. Use the (Eudoxus) Method of Exhaustion and sketch a proof that the ratio of the area swept out by one revolution of the spiral is 1/3 the area swept out by one area by the first circle.
3. Show that the Diophantine equation

   

   has no integer solutions, provided both m and n are integers.

4. Who first derived recursive relations used for the computation of ? What were these formulas? Derive them.
5. State and prove the general relationship between odd numbers and cubes shown below.

   

6. Explain what are epicycles, to what purpose they were used, and by whom.
7. Using plain geometry, prove the following: If a straight line is cut in extreme and mean ratio, the square on the sum of the lesser segment and half of the greater segment is five times the square on half of the greater segment.
8. Give a complete list of curves considered by ancient mathematician. Classify them as plane, conic, or higher, and explain your classification
9. Give two examples (with explanation) of how prevailing philosophical views hampered the development of ancient mathematics.
10. Outline the contents of all thirteen books of Euclid's Elements.

Due Date: March 17, 1997

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