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

Hippocrates’ Quadrature of the Lune, Part 2 Euclid’s Proof of the Pythagorean Theorem, Part 1

Question Set 2 Due: 9.10.2008

Chapter 1 (cont.)

√ √ √ 1. Construct the numbers 2, 3and 5 using a compass and an unmarked straight-edge. Briefly outline the steps followed in your constructions. √ 2. If a is a constructible number, then a is also constructible. √ (a) Explain how you would use a compass and unmarked straight-edge to construct a from a.

(b) The√ number π is transcendental. Use this fact together with the result you explained in (a) to show that π cannot possible be a constructible number.

1 3. Explain why two line segments of lengths 1 and 2 3 (same unit of measure) respectively are commensurable. 4. Statement: “a semi-circle is squarable.” Prove or disprove this statement using ideas from Chapter 1 in Dunham’s Journey through Genius. State precisely what results does your argument uses as they become relevant in your reasoning. 5. What is, in your opinion, the most surprising thing you have learned in this Chapter 1 of Dunham’s text? Describe in no less than half-a-page, your emphasis should be in the mathematics.

Chapter 2

1. Briefly describe how the following five proper names (of people and places) were connected: Socrates, Plato, Eudoxus, the Academy, the great Alexandrian Library, Euclid. 2. How were Eudoxus and Hippasus historically connected through mathematics? 3. Discuss how Eudoxus method of exhaustion can be seen as a “geometric forerunner of the modern notion of limit” used in Calculus. 4. The following item concerns Euclid’s Elements.

(a) How was this work physically organized? (b) Why is this work often regarded as an unprecedented contribution to mathematics? What are the merits of an axiomatic system such as Euclid’s? (c) Give a specific example of how The Elements has influenced Western thought outside of mathematics. (d) Characterize the following terms (appearing in Euclid’s Elements), emphasizing how they are different from each other: Definition, Postulate, Common Notion, Proposition, Proof.

5. Explain the meaning of the adjective “collapsible” in reference to a Euclidean compass. 6. Why does Euclid’s proof of the Pythagorean theorem deserve particular attention, even when this result had been proved before his time? What was the main strategy upon which Euclid built his proof?