Day 4, Mat305/594, History of Mathematics

Last time: Section 1.3

Next time: Sections 2.3

Today:

• Announcements
  • No class Monday (Labor day)
  • Worksheet for sections 2.3 and 2.4 (for next time). Note: this is a slight change from the assignment page, but since we have a week in front of us, it seems like you will be able to handle them both. Besides, many of the problems relate to the material of 2.2....
  • Note: your problems (for chapter 1) are due next Wednesday.

• Several things today:
  • Sherlock Holmes in Babylon -- conclusions
  • Problems/Comments from Section 1.3?
  • The Great Fraudini -- a sort of lame commercial...;)
  • Lessons from Sections 2.1/2.2

• Lessons and problems from section 1.3
  • What is the historical background you identified in section 1.3?

  • What are the geographic elements you identified in section 1.3?

  • Mathematical Elements?

  • Problems, pp. 29--, 2, 3, 5, 6, 10, 11, 13-16: any we should look at?

  • A collection of images of mathematics tablets (with explanations) -- beautiful stuff!

  • Sherlock Holmes in Babylon -- conclusions

• Lessons and problems from section 2.1
  • Note the quote at the beginning of the chapter, by Hermann Hankel:

    In most sciences one generation tears down what another has built and what one has established another undoes. In Mathematics alone each generation adds a new story to an old structure.

  • What is the historical background you identified in section 2.1?

  • What are the geographic elements you identified?

  • Mathematical Elements?

• The Great Fraudini

• Mathematical Elements, lessons and problems from section 2.2
  • Multiplication
  • Whole number division
  • When fractions arise
  • Multiplying with fractions
  • Dividing with fractions

• Napoleon's Campaign in Egypt
• Description de l'Egypte
• The Rosetta Stone:
  • An image of the stone
  • Its history and meaning
• Jean Francois Champollion and his grammaire Egyptienne en Encriture Hieroglyphique

• Ancient Egypt
• Thebes, Egypt

• Discovery of ancient Egyptian manuscripts
  • Participation of Gaspard Monge and Jean-Baptiste Fourier in the scientific expedition attached to Napoleon's army
  • Rhind papyrus (owned by A. Henry Rind, purporting (by Ahmes) to be mathematics of 1800 BC or so).
  • Moscow (or Golenischev) papyrus
    • Its discovery isn't too interesting.... This link gives its history, and a list of the 25 problems.
  • The Egyptian Mathematical Leather Roll (EMLR) - a paper elucidating

• From the Rhind: "Knowledge of all obscure secrets" comes down to multiplication and division.
• Multiplication is accomplished through binary decomposition (successive doublings).
• Unit fractions -- the key and the curse
• The Rhind papyrus contains the Table of unit fractions multiplied by 2, which is critical for doublings (this table constitutes 1/3 of the Rhind papyrus! -- p. 40).
• An example problem from the Moscow Papyrus
• What's with 2/3?
  

• The splitting identity --
  

  allows us to represent all rational numbers as sums of unit fractions.

• Fibonacci also developed a method for doing so.
• A fantastic site on Egyptian Fractions (including a calculator to turn your rationals into sums of unit fractions).
• Another
• Ancient Egyptian Science: A Source Book (by Marshall Clagett)

• The sad story of George Friedrich Grotefend (1775-1853)
  • Persian Cuneiform
  • 1802 -- results presented to the Academy of Science in Gottingen
• Henry Creswicke Rawlinson (1810-1895)
  • Behistun Inscription, erected by Darius the Great in 516 BCE, in Old Persian, Elamite, and Akkadian. Old Persian was cracked first.
  • Analysis of the Babylonian text at Behistun
  • "Thanks to Rawlinson's remarkable efforts... the vast records of Mesopotamian civilization were now an open book." (p. 23)
• A timeline that I'd like to develop, with your help. What I'll need from you are dates and descriptions of important moments missing from the timeline.

• Babylonian meaning "...the alluvial plain between the twin rivers, the Tigris and the Euphrates. The Greeks called this land 'Mesopotamia,' meaning 'the land between the rivers.'"
• Hammurabi's Babylonia, showing the Babylonian territory upon his ascension in 1792 BC and upon his death in 1750 BC
• Differences in geography between Babylon and Egypt meant that, whereas Egypt was safe from invasion, Babylon was overrun time and again! (Interesting dynamical system described on page 21).
• Tigris and Euphrates
• Gottingen (now Germany). "Some of the most famous mathematicians in history, Carl Friedrich Gauss, Bernhard Riemann and David Hilbert were professors at Gottingen."
• Persepolis
• Kermanshah Province, Iran, where Behistun is found.
• China and India

• Babylonian pictographs (about 3000 BC) turned into Cuneiform
• Technology implies limitations ("materials chosen for writing imposed special limitations of their own", p. 21)
• cuneiform = wedge
• Orientation of the wedge is important, implying that they understand the rudements of angles.
• Out of cuneiform arose the Babylonian positional number system
• sexagesimal system (confirmed in 1854 -- p. 24 -- Senkerah, on the Euphrates). Speculation on its origin:
  • Year has about 360 days
  • 60 has many divisors, making lots of fractions nice (terminate).
• Interesting that they used base 10 to make their set of 59 symbols making up their sexagesimal digits. (p. 24)
• Had subtraction sign (p. 24)
• Absence of zero meant that "position" was relative: "one could only rely on context to relieve the ambiguity." (p. 25)
• 300 BC, a "divider" was introduced, indicating an empty space -- used only medially and not at the end.
• "Babylonians of antiquity never achieved an absolute positional system." (p. 26)
  • In fact, our author claims that "The Babylonians were the only pre-Grecian people who made even a partial use of a positional number system." (p. 23), then contradicts himself on p. 27, re the Chinese.
  • The Mayan system is also called positional (#13, p. 30, and see p. 7), and they had a zero!
• "Fractions that had non-terminating sexagesimal expansions were approximated by finite ones...." (p. 26)
• Babylonian fractions beat Egyptian ones, because the Egyptians felt obliged to express fractions as sums of unit fractions (p. 26).
• Ancient Chinese sources don't exist, primarily due to differences in climate and writing materials (bamboo and cord, bark, or silk, p. 21 and 27).
• No mathematical texts survive from India, BCE (p. 27).
• Shih Huang-ti (China, 221 BC) tried to destroy all books of learning.
• There is a misconception that the Chinese developed a lot of important theoretical mathematics. Actually they were mostly focused on the calendar, and the claim is that, as of the 16th century, studying mathematics on one's own was forbidden under pain of death (p. 28).