1. Given that we have no documents from the time of Greek mathematics—the earliest manuscripts we have are medieval—how can we be sure that the texts we have are actually what the authors wrote? Were the copyists who wrote the early medieval manuscripts simply concerned with reproducing the text faithfully, or is it possible that they tried to improve it by revisions they thought of themselves? How could we know if they did?
2. What are the main topics investigated in ancient Greek number theory? How much of number theory has a practical application nowadays? (If you don’t know about RSA codes, for example, look them up on-line.)
3. What mathematical ideas were ascribed to the Pythagoreans by ancient commentators?
4. For what reason would the ancient Greeks have been investigating figurate numbers, perfect numbers, and the like? Did they have a practical application for these ideas?
5. Assuming that there are two square integers whose ratio is 5, derive a contradiction using the principle that underlies
Discuss the long-term implications of capital expenditures
Knorr’s conjecture. (If the integers are relatively prime, then both must be odd. Use that fact and the fact that the square of any odd number is one unit larger than a multiple of 8 to derive a contradiction.)
6. How do you resolve the paradoxes of Zeno?
7. Summarize the progress made on each of the three classical problems during the fourth century BCE.
8. Why was it important to Menaechmus’ solution of the problem of two mean proportionals that the plane cutting the cone be at right angles to one of its generators?
9. How would you establish that two triangles with equal altitudes and equal bases are equal (“in area,” as we would say, although Euclid would not)?
10. Why is the problem of squaring the circle much more difficult than the problem of doubling the cube or trisecting the angle?
11. It appears that the Greeks overlooked a simple point that might have led them to break out of the confining circle of Euclidean methods. If only they had realized that composite ratios represent multiplication, they would have been freed from the need for dimensional consistency, since their ratios were dimensionless. They could, for example, multiply and number of ratios, whereas interpreting the product of two lines as a rectangle precluded the possibility of any geometric interpretation of product containing more than three factors. Could they have developed analytic geometry if they had made this realization? What else would they have needed?
12. Granting that if two lines perpendicular to the same transversal line meet on one side of that line, reflection about the midpoint of the interval between the two points where the lines meet the transversal shows that they must also meet on the other side. How do you know that these two points of intersection are not the same point? What other assumption must you introduce in order to establish that they are different?
13. Was the Elements an exposition of the most advanced mathematics of its time?
14.What advances in geometry, beyond the basic results found in the Elements, are due to Archimedes?
15. Show from Apollonius’ definition of the foci that the product of the distances from each focus to the ends of the major axis of an ellipse equals the square on half of the minor axis.
Mathematics
