Day 13, Mat305/594, History of Mathematics

Last time: Section 3.5

Next time: Section 4.3

Today:

• Announcements
  • Homework through section 3.2 due.
  • Some beautiful "behold proofs" of a few infinite series
  • Next time: we may well have a visit from Frank Morgan.
  • I've not prepared a worksheet for 4.3. It's a collection of number theoretic results, and we'll pick some of the greatest hits (e.g. the infinitude of primes, Euclid's division algorithm) to concentrate on. Homework: problems pp. 184, #1, 6, 10, 15, 18, 20, 26.

• Today:
  • Lessons from 4.2
  • Commercial (Paul or Debra?) for Wednesday; the other for next time.
  • More Lessons from 4.2

• Mathematical Elements, lessons and problems from section 4.2: "Here's lookin' at Euclid"
  • Robin Wilson in "Lewis Carroll in Numberland"

    • Lewis Carroll liked to have fun in all things:

      "If AB were to be divided into two parts at C..."
      "It would be drownded."

    • Sorry if I offended anyone with my editorial comment about "Our educational system": but I too find that "... studies may be made so easy and mechanical as to render thought almost superfluous." I hope that our course is not so!
    • 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....
    • 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?

  • Another resource for Euclid's propositions and proofs

  • Euclid's Elements
    • "Proclus ... speaks of all of mathematics as hypothetical; that is, it merely deduces what must follow from the assumptions, whether or not the latter are true." p. 59. This is a very mature mathematical attitude. Proclus (8 February 412 - 17 April 485 AD) is the one who provides us our single element of information about Euclid's life -- about 750 years after the fact! What could you tell with any confidence about those who lived in 1250 AD?

    • Book I -- Elementary Plane Geometry
      • Openings:
        • Definitions

        • Postulates

          (especially the parallel postulate)

        • Common Notions

          truths applicable to all sciences

      • Proposition 1 -- we've already done!
      • Proposition 5 -- "the asses' bridge"
      • Proposition 47 -- "the windmill"
      • Proposition 48 -- the converse of the Pythagorean Theorem

    • Book II: Geometric Algebra
      • In which equations are solved by geometric construction
      • The golden section (p. 166). Our author's construction is much nicer than that found in Book II, Proposition 11

    • Construction of regular n-gons
      • Greeks resolved their fear of "the unutterable" by identifying irrational numbers with geometric quantities ("Main theme of the section" (p. 169))