Day 16 in Mat115H

Today:

• Announcement:
  • Exam 1 is graded
    • Although there were 110 points possible, I'm treating your total score as a percentage (which means that four of you have scores over 100%).
    • Mean: 85.1
    • Median: 87.75
    • Standard Deviation: 16.6
    • Spelling!
      "received" was misspelled on at least four papers (i before e, except after c).
  • If you have particular issues, feel free to come and see me.

• More Mathematics of Fit the Fourth
  • Four Impossible Problems (remember the White Queen!)
    1. "squaring the circle" -- constructing a square with the same area as a given circle.

       I heard this one on Hardball with Chris Matthews last night! "How are they going to square that circle?" he asked.

    2. "doubling the cube" -- constructing a cube with twice the volume of a given cube.
    3. trisecting an angle -- cutting a given angle into three equal angles.
    4. Deriving the parallel postulate from the other four postulates (and common notions, etc.)

  • Skills:
    1. Bisecting a given segment
    2. Dropping a perpendicular to a given line from a given point
    3. Bisecting a given angle
    4. Drawing a parallel to a given line through a given point.

  • Together:
    • a useful version of Euclid
    • Constructing a regular triangle (equilateral triangle): Proposition I.1 of Euclid
    • Constructing a regular hexagon (see p. 28)
    • Trisecting a 90^o angle (see "homework", p. 28)
    • Proof of the asses' bridge (p. 84), Proposition I.5. What can we assume?
      • Postulates, definitions, common notions
      • Proposition I.4 (side-angle-side)
    • Proof of the Pythagorean Theorem (p. 82)
      • Euclid's proof (Proposition I.47)
      • we'll do an easier proof than that!

  • The Euclid Debate: "De Morgan raised the question whether Euclid be, as many suppose, the best elementary treatise on geometry, or whether it be a mockery, delusion, snare, hindrance, pitfall, shoal, shallow, and snake in the grass." -- An Anti-Euclid Association was formed....

  • Dodgson ably defended Euclid, but could not triumph over all the arrayed forces. What's Dodgson's point in the Minos sketch on page 93?

  • What do you think of Dodgson's alternative postulate to the parallel postulate (p. 98)?

  • How about his approach to attacking circle squarers?