Math 305, Section 1 • Mathematics from a Historical Perspective • UNM, Fall 2008

Hippocrates’ Quadrature of the Lune Part 1

Question Set 1 Due: 9.3.2008

1. What is the term “awakening” used to describe in relation to the history of mathematics? 2. What were the two main contributions of the Pythagorean brotherhood to modern mathematics? 3. What does it mean for two quantities to be commensurable? 4. Hippasus discovered that the side of square and its diagonal are not commensurable. How does this amount to the discovery of irrational numbers, speaking in today’s terms? 5. Why was Hippasus discovery regarded as a triumph of geometry over arithmetic? 6. What is it meant by the word “quadrature,” and why were the Greeks concerned with finding out which figures were quadrable? 7. Draw a compass and straight-edge picture that shows a Lune known to be quadrable today. 8. Using a compass and a straight-edge only, construct a hexagon inscribed in a circle. Describe, using a detailed series of steps, the construction in question. 9. Hippocrates of Chios, Lindemann, Tschebatorew, and Dorodnow each contributed to the results known today concerning quadratures of Lunes. Explain what each of these individuals/collaborators did and how it built on and/or formed a foundation for the work of the others. 10. What is a constructible number? Give an example of a number that is constructible, and explain how you know it is. Give an example of a number that is not constructible and explain how you know it is not. 11. State Lindemann’s result on the transcendental nature of π. Explain how this result can be used to prove that the circle is not squarable. 12. Write a summary of Chapter 1 of Dunham’s book that fits in at most two (single-sided) pages. The emphasis should be on describing how the different results presented weave a piece of mathematical history through time.