Islamic Mathematics, an Overview MATH 110-2 – Algebra Through History October, 2019

1. Islamic Mathematics, an OverviewMATH 110-2 – Algebra Through HistoryOctober, 2019

2. The key role of Baghdad • The Greek approach to deductive mathematics (a la Euclid, not so much Diophantos) was extremely influential for later developments in the subject • But it was not the only important strand • One reason that things developed this way: many of the Greek mathematical texts we have discussed were translated into Arabic 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 also not at all completely “pure mathematics” 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. 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. • Additional works extending some of the number-theoretic sections of the Elements

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

9. 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.

10. Omar Khayyam Lived ca. 1040 – 1123 CE, in Persia (Iran); not associated with milieu of 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

11. Question: Was the Islamic role just “transmission?” Quotes from a (controversial) historian named 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.”

12. G. Joseph's model for the history of math during the “Dark Ages” (see his book “The Crest of the Peacock”)