Recall G. Joseph's model for the history of math during the

1
Recall G. Joseph's model for the history of math
during the Dark Ages

2
The key role of Baghdad

• As we have said, the Greek approach to deductive
  mathematics (a la Euclid) has been extremely
  influential for later developments in the subject
• But it's not the only important strand
• One reason that things developed this way many
  of the Greek mathematical texts we have discussed
  were preserved and studied in the Bayt-al-Hikma
  (House of Wisdom) in Baghdad during the Abbasid
  period, 750 1258 CE

3
The key role of Baghdad

• Text of works of Euclid obtained about 800 CE by
  way of Byzantine empire (under caliph Harun
  al-Rashid)
• Claudius Ptolemy's Mathematike Syntaxis
  Almagest translated into Arabic in 827 CE
• Also translated Aristotle, Apollonius (conic
  sections), Archimedes, Heron, many other Greek
  works some survive only in this form
• In addition, key Indian texts were also
  translated into Arabic here

4
Some key players in this story

• Muhammad ibn Musa al-Khwarizmi (ca. 780 850 CE)
  
• Name suggests he or his family came from a region
  in current-day Uzbekistan (north-east of Iran)
• Invited to come to Baghdad about 820 during reign
  of caliph al-Mamun
• Spent the rest of his life there under the
  patronage of the caliph and his successors

5
al-Khwarizmi

• Probably his most famous book Hisab al-jabr
  w'al muqabala Compendium on calculation by
  restoration and reduction
• Does al-jabr sound familiar? (It should, if
  you think about it!)
• Gave general methods for solving quadratic
  equations, other types of solving methods and
  manipulations, beyond any previous work we know
  of in some cases
• Doesn't use symbolic expressions, though all
  expressed verbally and/or geometrically

6
al-Khwarizmi

• The Hisab al-jabr w'al muqabala was not at all
  completely pure mathematics, though
• Also an extensive section on solving problems
  about questions of distribution of bequests in
  wills and inheritances (a big subject in Islamic
  law)
• Involves pretty extensive and intricate
  computations with fractions(!)

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al-Khwarizmi

• Arguably, his most influential work, though, was
  a book that survives only in a Latin translation
  Algorithmi de numero indorum (Calculation with
  Indian numerals, i.e. what are often called
  Hindu-Arabic numerals today)
• This work popularized the use of the base-10
  positional number system, with symbol for 0, that
  we still use today!
• Our modern word algorithm step-by-step process
  for solving a problem comes from al-Khwarizmi's
  name(!)

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Hindu-Arabic numerals

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Thabit ibn-Qurra

• From northern Mesopotamia, lived ca. 836 901 CE
• Moved to Baghdad as an adult and joined the
  translators working on mathematical texts in the
  House of Wisdom
• Wrote a work extending the Quadrature of the
  Parabola of Archimedes (so at least some people
  were reading Archimedes with understanding!)

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Thabit ibn-Qurra

• Additional works extending some of the
  number-theoretic sections of the Elements
• We'll look at this in some detail next time,
  because the story is a fascinating one
• But we need two be able to write some formulas
  for this, so we'll switch to a different method
  for making slides(!)

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Thabit ibn-Qurra

• Also made a critical re-examination of the basis
  of the Elements, including a serious attempt to
  prove Postulate 5 from the other Postulates and
  Common Notions
• Postulate 5 If the two interior angles on one
  side of a transversal to two lines add to less
  than two right angles, then the two lines, if
  extended indefinitely, will meet on that side of
  the transversal.

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Thabit also gave a dissection proof of the
Pythagorean theorem equivalent to the Chinese
go-gou construction
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Thabit ibn-Qurra

• An interesting question here Did he have access
  to Chinese sources? (Or was knowledge of the
  cut and paste geometry from the Babylonian
  period still preserved?)
• Tempting to speculate, but no firm evidence
  either way
• There were trade and other more or less indirect
  contacts between the Islamic caliphate and China
  by way of India, so it's not out of the question.
  

14
Omar Khayyam

• Lived ca. 1040 1123 CE, in Persia (Iran) not
  associated with the House of Wisdom in Baghdad
• Known both as a poet and as a mathematician,
  astronomer, and philosopher
• Biggest mathematical contribution were algebraic
  and geometrical methods for solving various sorts
  of cubic and higher-degree equations
• Definitely went beyond the Greeks here

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Omar Khayyam

• Also known for a book called Explanations of the
  difficulties in the postulates in Euclid's
  Elements.
• The book consists of several sections, one on the
  parallel postulate (Book I), one on the Euclidean
  definition of ratios (later books) and others.
• One of the first to have the idea of using an
  opposite form of the 5th Postulate and trying
  to reason to a contradiction.
• This work was inconclusive, though.

16
Question Was the Islamic role just
transmission?

• More quotes from our favorite whipping boy
  Morris Kline
• The significant contribution to mathematics that
  we owe to the Arabs was to absorb Greek and Hindu
  mathematics, preserve it, and ultimately, ,
  transmit it to Europe.
• The Arabs did make critical commentaries of
  Euclid's Elements, which is surprising because it
  shows appreciation of rigor despite their usual
  indifference to it in algebra.