Math 464, Exam 1
Katz (Ch. 1-6), Dunham (Ch. 1-4)

To be able to Prove or Explain:

• Katz 1.5:

  • How the Egyptians may have found their approximation to the area of a circle, , where is its diameter. (pp.20-21)

  • How the Babylonians and Chinese may have found their approximation to the area of a circle, , where is its diameter. (p.21)

  • How the Babylonians and Chinese may have found the correct formula for the area of a circle, , where is circumference and diameter. (Two different methods were shown.) (pp.20-21)

• Katz 1.8: The Chinese geometric solution to problem #6 in the Jiu Zhang (the reed in the pond). (p.33)

• Katz 2.1.1: The incommensurability of the side and diagonal of a square. (p.51)

• Dunham Chap. 1: Two Preliminaries (Triangle to Rectangle, to Square), followed by Great Theorem and Proof 1: Hippocrates of Chios Squares the Lune

• Katz 2.4.2:

  • Proof: Euclid I-6 (p.65)

  • Proof: Euclid I-29, Euclid's first use of Postulate 5 (p.66)

  • Proof: Euclid I-32, Angle sum is two right angles (Ex.11, p.96)

• Dunham Chap. 2:

  • One Preliminary (I-41), followed by Great Theorem and Proof 2: Euclid's Proof of I-47, the Pythagorean Theorem

  • Proof of I-48, the Pythagorean Theorem Converse

• Katz 2.4.3: Proof: II-11 (p.70)

• Katz 2.4.6: Proof: VI-12, "Construct , so that " (Class notes, & p.83)

• Dunham Chap. 3: Preliminaries (Defs of prime and composite), following by Great Theorem and Proof 3: Euclid's Proof of IX-20, infinitude of primes

• Katz 2.4.9: Proof: XII-2, Circles to diameters-squared (Method of Exhaustion) (pp.91-92)

• Katz Addendum: Eratosthenes, Circumference of the Earth (Class notes, & Ex.41, p.99)

• Katz 3.1.1: Archimedes "On the Equilibrium of Planes".

  • Proof: Proposition 1 (p.105)

  • Proof: Proposition 6, The Law of the Lever (p.106) (Commensurable case only)

• Dunham Chap. 4: Four Preliminaries (See Class notes), followed by Great Theorem and Proof 4: Archimedes "On the Measurement of The Circle", Proposition 1, The Area of a Circle

• Katz 3.3.1: Archimedes "The Method", Proposition 1, Area of a Parabolic Segment

  • Ideas of 1st Proof: By the Law of the Lever (pp.111-112)

  • Ideas of 2nd Proof: By the Sum of Series (pp.113-114)

• Katz 3.5.1: Apollonius Proposition I-33, drawing a tangent to a parabola (pp.123-124)

• Katz 3.5.3: Archimedes' Trisection of an Angle with a marked straightedge (Class notes)

• Katz 4.2.1: Ptolemy's Theorem and Proof (p.148)

• Katz 5.2.1: Diophantus Arithmetica, II-13 (Class notes, & near p.177)

• Katz 6.3.1: Chinese Remainder Problem and solution (pp.197-198)

To be able to State, Understand, or Use:

• Katz 2.1.2: The three famous Greek construction problems and the restrictions under which they were to be solved. (pp.51-52)

• Katz 2.3.2: The Greek distinction between Numbers (The Discrete) and Magnitude (The Continuous), and its implications (pp.56-57)

• Katz 2.4.1: Statement & Explanation: Euclid's five postulates (p.61). Statement: Euclid's five common notions (p.62)

• Katz 2.4.3: Statement: II-4, Example of Geometric Algebra (p.68)

• Katz 2.4.5: Statement & Use: Euclidean Algorithm, gcd of two numbers (pp.78-79)

• Katz 2.4.7: Statement & Translation: IX-35, Partial sums of geometric sequences (pp.87-88)

• Katz 2.4.8:

  • Statement: Euclid X-1 (p.89)

  • Statement: Euclid X-9 (p.89)

• Katz 3.1.1: Archimedes "On the Equilibrium of Planes": Statements: Postulate 1 [Principle of Insufficient Reason], and Postulates 2-3 (p.105)

• Katz 3.2: Quadratrix of Hippias, description (p.110); Use in trisecting an angle. (Class notes)

Other Questions

• Proof Explication of some not previously seen result