Day 20, Mat305/594, History of Mathematics

Last time: Section 4.5

Next time: Sections 5.3 and 5.4

Today:

• Announcements
  • Sorry, I haven't graded your homework yet. Hopefully by Monday.
  • The remainder of your exams are graded. A few of your history art pictures:
    • Egyptian (best of show)
    • Babylonian
    • Pythagorean
  • Worksheet for section 5.3
  • We'll also try to cover 5.4 along with 5.3 next time, hitting the highlights (especially Hypatia!).
  • Let's make the homework through 4.3 due next Wednesday.

• Today:
  • A last salvo from Archimedes
  • Commercial from Katie; then Nathan, next Monday. Who's next?
  • Lessons from 5.1/5.2

• Mathematical Elements, lessons and problems from section 4.5: Archimedes and Apollonius
  • Archimedes of Syracuse
    • Born, lived and died in Syracuse. Father Phidias an astronomer.
    • "greatest creative genius of the ancient world"
    • Of the royal family of King Hieron II
    • Visited Egypt, and presumably knew the mathematicians of the Museum in Alexandria (e.g. Eratosthenes)
  • Images of Archimedes
  • Archimedes's Mathematics
    • A last salvo: The Quadrature of the Parabola
    • Mathematica Code, in which we do a little calculus.
    • Source of Archimedes's approach
    • Source of Archimedes's geometric series solution of the area

• Mathematical Elements, lessons and problems from sections 5.1/5.2: Twilight of the Greeks
  • Twilight of the Greeks
    • The Museum heads into decline at the end of the 3rd Century. In 146 BC Ptolemy VII sends the scholars and scientists who did not support him into exile -- this does nothing other than enrich the neighboring countries, while impoverishing his own realm.
    • Diophantus brings a last, brief, bit of fame to the Museum.
    • The Romans begin their accession (e.g. the Battle of Syracuse)
      • Greece falls by 146 BC
      • Mesopotamia falls by 64 BC.
      • Romans in turmoil by 44 BC, with the assassination of Julius Caesar.
      • 30 BC, Octavian enters Alexandria in triumph.
      • The Roman Empire, circa AD 117, corresponding to the period that Gibbon characterized thus: "If a man were called to fix the period in the history of the world during which the condition of the human race was most happy and properous, he would, without hesitation, name that which elapsed from the death of Domition to the accession of Commodus (96-180 A.D.)."
      • Overall picture: declining mathematical activity and originality. The Romans were engineers (viaducts, colliseums), not theoretical mathematicians.
    • Christianity contributes to the decline of Greece. Theological pursuits displace scientific studies. Mobs destroy libraries. The Museum comes to an end.
    • The Roman empire ends, fading out as hordes pick it apart.
    • Constantinople, center of the eastern end of the Roman empire (the Byzantine empire), keeps the lights on: "a knowledge of the Alexandrian tradition never completely died out in the East...."

      "Without the efforts of the Byzantine copyists, most of the ancient scientific and literary texts would have been lost forever. There might never have been a Renaissance." (p. 219) (!)

  • Diophantus
    • Life details few, although we have this from the epigram at his grave (referenced, p. 220): "This tomb holds Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life: when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! Late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life." Metrodorus

      From this, we can solve to find that his boyhood lasted 14 years, his beard grew at 21, he married at 33; a son was born to him five years later and died, at the age of 42, when Diophantus was 80 years old. Diophantus's own death followed four years later.

      Our author states that he was most likely a Hellenized Babylonian.

    • Arithmetica his great work in algebra: only 6 of 13 books remain, 189 problems, with solutions in integers and rational numbers.
    • Fermat wrote his famous margin comment in the Arithmetica ("Fermat's last theorem").
    • Negatives were not allowed: the equation 4x+20=4 was "absurd" since it would lead to the impossible solution x=-4.
    • Diophantus had to invent his own notation, to "emancipate algebra" from its geometric chains
      • He introduced a variable (the unknown)
      • powers to the sixth
      • equality
      • subtraction
      • reciprocals
      • Interestingly, he didn't use a sign for addition (just appending things to add); all terms with negative coefficients had to be placed after a single subtraction sign. See the equation at the top of page 222.
      • He only had one symbol for the unknown, so "he was compelled to reduce all his problems, no matter how complicated, to equations in one unknown."
    • The apparent object of the Arithmetica "was to teach the method of solution of certain problems in which it is required to find rational numbers satisfying prescribed conditions." (p. 221)

      Example problems from the text (pp. 222-):

      1. Use: 30, 29, 25, 21
      2. Use:
      3. Use:
      4. Use: 8 and 9
      5. Use: x and 9x-1 (this requires a change at the choice of factorization of x! What is it?)
  • A good working reference on Diophantus