MATH 380 Assignment #2

1. Find out what you can about the life of Thales of Miletus (start with Section 4.4.1 of the text) or Pythagoras of Samos (start with Section 4.4.2), and our sources of information about him. How reliable, in your opinion, is the data, especially that about his education and his work? Why? [5]

2. Thales is supposed to have shown that an angle inscribed in a semicircle is a right angle (see Section 4.4.1). Another way of putting this is to say that if a triangle ABC is inscribed in a circle and side AB is a diameter of the circle, then the angle at C is a right angle. Assuming, if you wish, that Thales' theorem is true, prove that its converse, given below, holds as well. [10]

If a triangle ABC is inscribed in a circle and the angle at C is a right angle, then side AB is a diameter of the circle.

3. Write a short essay -- no more than 500 words -- about {\it one\/} of the following three topics. [10]

• The philosopher Plato held that material reality is but a shadow of a fundamental reality of timeless and unchanging "forms". How was Plato's concept influenced by Pythagoras' conception of numbers as the fundamental reality?

• Pythagoras is also supposed to have to done some work on mathematics related to music (see Exercise 4.13). Describe this (alleged) work.

• Describe how modern mathematics could deal with Zeno's Achilles paradox (see Section 4.4.3) and discuss to what extent (and why) such an approach could have been intelligible to Zeno and his contemporaries.

4. Exercise 4.5 [5]

Bonus. In Exercise 4.7 it is asserted that Hypatia was "the only woman mathematician of ancient times whose name is recorded by the standard sources." Is this assertion true? Why or why not? [2]

Note: Please be nice to your fellow classmates and do not take any relevant books out of Bata Library. Replace them on the shelves, exactly where you found them!

Department of Mathematics

Trent University