Day 15, Mat305/594, History of Mathematics

Last time: Visit by Frank Morgan

Next time: Section 4.4

Today:

• Announcements
  • Wasn't that a great visit from Frank Morgan?
    I don't have a lot of quotes about the calculus of variations, but here's a particularly relevant one:

    Charles Dodgson (Lewis Carroll) -- written in his diary as a student:

    "I talked over the 'Calculus of Variations' with Price today, but without any effect. I see no prospect of understanding the subject at all."

  • Did you understand his "illogical" bit?

• Today:
  • Lessons from 4.2
  • Commercial (Debra); Paul for next time (next Wednesday); Ashley for the Monday following
  • Lessons from 4.2/4.3
  • Distribute Midterm
    • It's your own work!
    • If you have questions regarding the midterm, the answers may be found here. If not, feel free to ask!
    • I'll only be around until Friday at 3:00, however; then I'll be back late Monday night. I won't be available over the weekend. Get your questions in early!

• Mathematical Elements, lessons and problems from section 4.2: "Here's lookin' at Euclid"
  • 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). Kepler called this "one of the two Jewels of Geometry"

        Our author's construction is much nicer than that found in Book II, Proposition 11

    • Construction of regular n-gons
      • "Much of the history of classical mathematics could be written around the idea of the golden section." (p. 167) Wow!
      • Notice that the key to creating the regular decagon is to cut a line in golden section (p. 167)
      • Greeks resolved their fear of "the unutterable" by identifying irrational numbers with geometric quantities ("Main theme of the section" (p. 169))

• Mathematical Elements, lessons and problems from section 4.3: Euclid's Number Theory
  • "Euclid was particularly interested in questions pertaining to divisibility." (p. 172)
  • Euclid's division algorithm (VII.1, VII.2)
  • If a prime divides a product, then it divides one of the factors (VII.30)
  • Fundamental Theorem of Arithmetic (IX.14)
  • The infinitude of primes (IX.20)
  • The sum of a geometric series (IX.35)