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(!)

7. 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(!)

8. Hindu-Arabic numerals

9. 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!)

10. 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(!)

11. 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.

12. Thabit also gave a dissection proof of the Pythagorean theorem equivalent to the Chinese “go-gou” construction

13. 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

15. 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 5 th 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.”